Pengguna:Klasüo/bak pasir

Arsip 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 11, 12, 13, 14, 15, 16, 17, 18, 19, 20 21, 22, 23, 24, 25, 26, 27, 28, 29, 30 31, 32, 33

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Logaritma
Domain dan Citra
Domain dari fungsi	$(0,\infty )$
Daerah hasil fungsi	$(-\infty ,\infty )$
Nilai-nilai spesifik
Nilai di $+\infty$	$\infty$
Nilai maksimum	Tidak ada
Nilai minimum	Tidak ada
Sifat khusus
Akar	$1$
Invers	$x=b^{y}$
Turunan	${\frac {1}{x\ln b}}$
Antiturunan	$x\log _{b}x-{\frac {x}{\ln b}}+C$

Plot fungsi logaritma, dengan tiga basis yang umum digunakan. Titik-titik khusus log_b b = 1 ditandai dengan garis bertitik-titik, dan semua irisan kurva pada log_b 1 = 0.

Dalam matematika, logaritma adalah fungsi invers dengan eksponen. Itu berarti logaritma dari suatu bilangan yang diberikan x adalah eksponen dimana bilangan tetap lainnya, basis b dimasukkan untuk menghasilkan bilangan tersebut, yaitu x. Dalam kasus yang sederhana, logaritma menghitung jumlah kemunculan faktor yang sama dalam perkalian berulang; misalnya karena 1000 = 10 × 10 × 10 = 10³, "basis logaritma 10" dari 1000 adalah 3, atau log₁₀ (1000) = 3. Logaritma x ke basis b dinotasikan sebagai log_b (x), atau tanpa tanda kurung log_b x, bahkan tanpa basis eksplisit log x, ketika tidak ada kebingungan yang mungkin terjadi, atau ketika basis tidak penting seperti dalam notasi O besar.

The logarithm base 10 (that is b = 10) is called the decimal or common logarithm and is commonly used in science and engineering. The natural logarithm has the number e (that is b ≈ 2.718) as its base; its use is widespread in mathematics and physics, because of its simpler integral and derivative. The binary logarithm uses base 2 (that is b = 2) and is frequently used in computer science.

Logarithms were introduced by John Napier in 1614 as a means of simplifying calculations. They were rapidly adopted by navigators, scientists, engineers, surveyors and others to perform high-accuracy computations more easily. Using logarithm tables, tedious multi-digit multiplication steps can be replaced by table look-ups and simpler addition. This is possible because of the fact—important in its own right—that the logarithm of a product is the sum of the logarithms of the factors:

$\log _{b}(xy)=\log _{b}x+\log _{b}y,$

provided that b, x and y are all positive and b ≠ 1. The slide rule, also based on logarithms, allows quick calculations without tables, but at lower precision. The present-day notion of logarithms comes from Leonhard Euler, who connected them to the exponential function in the 18th century, and who also introduced the letter e as the base of natural logarithms.

Logarithmic scales reduce wide-ranging quantities to smaller scopes. For example, the decibel (dB) is a unit used to express ratio as logarithms, mostly for signal power and amplitude (of which sound pressure is a common example). In chemistry, pH is a logarithmic measure for the acidity of an aqueous solution. Logarithms are commonplace in scientific formulae, and in measurements of the complexity of algorithms and of geometric objects called fractals. They help to describe frequency ratios of musical intervals, appear in formulas counting prime numbers or approximating factorials, inform some models in psychophysics, and can aid in forensic accounting.

In the same way as the logarithm reverses exponentiation, the complex logarithm is the inverse function of the exponential function, whether applied to real numbers or complex numbers. The modular discrete logarithm is another variant; it has uses in public-key cryptography.

Motivasi

grafik dari basis logaritma 2 memotong sumbu-x antara x = 1 dan melewati titik (2, 1), (4, 2), dan (8, 3) yang menggambarkan log₂(8) = 3 dan 2³ = 8. Grafik secara arbiter mendekati sumbu-y, tetapi tak-memenuhinya.

Penjumlahan, perkalian, dan eksponen adalah tiga dari operasi aritmetika yang paling mendasar. Kebalikan dari penjumlahan adalah pengurangan, dan kebalikan dari perkalian adalah pembagian. Sehingga, logaritma adalah operasi kebalikan dari eksponen. Eksponen adalah ketika suatu bilangan b, basis dipangkatkan ke pangkat tertentu y, eksponen untuk memberikan nilai x; dilambangkan

$b^{y}=x.$

Misalnya, dipangkatkan 2 ke pangkat 3 menghasilkan 8, adalah: $2^{3}=8$

Logaritma basis b adalah operasi kebalikan, yang memberikan keluaran y dari masukan x. Artinya, $y=\log _{b}x$ sama dengan $x=b^{y}$ jika b adalah bilangan real positif. (Jika b dalam bukan bilangan real positif, eksponensial dan logaritma didefinisikan, tetapi mungkin memerlukan beberapa nilai, yang membuat definisi menjadi lebih rumit.)

Salah satu motivasi historis utama memperkenalkan rumus logaritma adalah

$\log _{b}(xy)=\log _{b}x+\log _{b}y,$

yang memungkinkan (sebelum ditemukannya komputer) untuk mengurangi perhitungan perkalian dan pembagian menjadi penambahan, pengurangan dan pencarian tabel logaritma.

Definisi

Logaritma dapat dipahami secara matematis sebagai anggota fungsi (yang keseluruhannya dilambangkan dengan $b$) dari $\mathbb {R} ^{+}\to \mathbb {R}$. Fungsi logaritma masing-masing hanya berbeda (tetapi tidak sama dengan nol) kelipatan satu sama lain.

Ini diperkenalkan dengan cara yang berbeda pada bilangan real positif. Tergantung pada latar belakang dan niat, Anda akan memilih satu atau pendekatan didaktik lainnya. Definisi yang berbeda dari logaritma real antara satu sama lain ekuivalen dan dibuat dengan fokus khusus pada logaritma alami, yang terjadi secara alami dari sudut pandang para matematikawan, seperti yang dilihat dalam akses melalui integral tak tentu dari ${\tfrac {1}{t}}$.

Sebagai fungsi invers dari fungsi eksponensial

Logaritma ke basis $b$ adalah fungsi invers dari fungsi eksponensial pada basis positif $b$, yaitu:

$x\mapsto b^{x}.$

Oleh karena itu, fungsi $b^{x}$ dan $\log _{b}x$ adalah fungsi invers satu sama lain. Logaritma membalikkan eksponensial dan sebaliknya:

$b^{\log _{b}x}=x\quad {\text{dan}}\quad \log _{b}(b^{x})=x.$

Hasil logaritma alami dari basis $b=\mathrm {e}$, dimana

$\mathrm {e} =2{,}718281828459\ldots$

yang adalah bilangan Euler.

Sebagai solusi untuk persamaan fungsional

Fungsi logaritma adalah solusi kontinu non-trivial $L$ dari persamaan fungsional

$L(x\cdot y)=L(x)+L(y).$

Solusi dapat memenuhi $L(1)=0$ dan bahkan dapat diturunkan. Logaritma alami kemudian diperoleh secara bersama dengan kondisi tambahan

$L'(1)=1.$

Kondisi tambahan adalah salah satu alasan untuk sebutan logaritma yang diperoleh secara alami. Jikalau mendapatkan logaritma ke basis lain $b$ melalui kondisi tambahan, maka yang harus dilakukan adalah

$L'(1)={\frac {1}{\ln b}}$

dan selalu membutuhkan logaritma alami lagi.

Solusi trivial dari persamaan fungsional atas adalah fungsi null $L(x)=0$, yang tidak dianggap sebagai fungsi logaritmik, dan satu-satunya solusi dari persamaan fungsional yang juga didefinisikan $L(0)$.

Karena persamaan fungsional atas, maka dari itu logaritma menyampaikan khususnya pemetaan pelestarian struktur dari bilangan real positif dengan struktur perkalian. Apabila seseorang juga menemukan secara eksplisit ini sebagai suatu kondisi dan dengan demikian sampai pada turunan.

Sebagai isomorfisme

Logaritma bernilai real adalah kontinu dari isomorfisme

$L\colon (\mathbb {R} ^{+}\!,\,\cdot \,)\longrightarrow (\mathbb {R} ,+)$.

Definisi ini dengan jelas mendefinisikan fungsi $L$ kecuali untuk konstanta perkalian.

Pendekatan aljabar menekankan seperti pendekatan melalui persamaan fungsional signifikansi secara historis logaritma sebagai bantuan perhitungan: Ini memungkinkan perkalian untuk "dikonversi" menjadi penjumlahan.

Sebagai antiturunan dari f dengan f(x)=1/x

Logaritma alami sebagai luas bawah grafik 1/x

Fungsinya adalah

$L\colon t\mapsto \int _{1}^{t}{\frac {1}{x}}\,\mathrm {d} x$

dengan $t>0$ adalah logaritma alami: Ini adalah $L=\ln$. Logaritma dengan basis $b$ diperoleh dengan membagi fungsi $L$ dengan konstanta $L(b)=\ln b$. Sebagai integral tak wajar dari $f$, atau batas integrasi bawah arbiter (positif), dianggap hanya akan mendapatkan satu tambahan, konstanta aditif, tetapi hanya mendapatkan logaritma ke basis $\mathrm {e}$.

Sebagai deret pangkat

Logaritma alami dapat dinyatakan sebagai deret pangkat, yaitu

$\ln(1+x)=\sum _{k=1}^{\infty }(-1)^{k+1}{\frac {x^{k}}{k}}=x-{\frac {x^{2}}{2}}+{\frac {x^{3}}{3}}-{\frac {x^{4}}{4}}+\dotsb$

Deret ini adalah konvergensi untuk $|x|<1$ dan untuk $x=1$.

Untuk perhitungan numerik dari nilai $\ln(1+x)$ untuk $x>1$ pada relasi $\ln(1+x)=-\ln {\Bigl (}1+{\Bigl (}-{\frac {x}{1+x}}{\Bigr )}{\Bigr )}$ yang digunakan.

Catatan

Definisi ini juga dapat digunakan untuk mendapatkan logaritma pada struktur matematika lainnya, seperti: sebuah bilangan pada bilangan kompleks. Ini mengasumsikan bahwa konsep yang digunakan untuk definisi ada dalam struktur yang bersangkutan.

Misalnya, untuk mendefinisikan logaritma diskret pada grup, Konsep seperti diferensiasi/integrasi tidak dapat digunakan karena mereka bahkan tidak digunakan. Definisi terjadi karena sebagai inversi eksponen dengan seluruh eksponen, yang pada gilirannya didefinisikan oleh beberapa penggunaan pranala satu grup.

Identitas logaritmik

Artikel utama: Daftar identitas logaritma

Beberapa rumus penting, terkadang disebut identitas logaritmik atau hukum logaritma, menghubungkan logaritma satu sama lain.

Darab, hasil bagi, kuasa (pangkat), dan akar

Logaritma dari sebuah darab adalah jumlah logaritma dari bilangan-bilangan tersebut dikalikan; logaritma dari rasio dua bilangan adalah perbedaan dari logaritma. Logaritma dari sebuah kuasa (pangkat) ke-p dari suatu bilangan adalah p dikali logaritma dari bilangan itu sendiri; logaritma dari akar ke-p adalah logaritma dari bilangan dibagi p. Tabel berikut mencantumkan identitas ini dengan contoh. Setiap identitas dapat diturunkan setelah substitusi definisi logaritma $x=b^{\,\log _{b}x}$ atau $y=b^{\,\log _{b}y}$ di sisi kiri.

	Rumus	Contoh
Darab	${\textstyle \log _{b}(xy)=\log _{b}x+\log _{b}y}$	${\textstyle \log _{3}243=\log _{3}(9\cdot 27)=\log _{3}9+\log _{3}27=2+3=5}$
Hasil bagi	${\textstyle \log _{b}\!{\frac {x}{y}}=\log _{b}x-\log _{b}y}$	${\textstyle \log _{2}16=\log _{2}\!{\frac {64}{4}}=\log _{2}64-\log _{2}4=6-2=4}$
Kuasa (pangkat)	${\textstyle \log _{b}\left(x^{p}\right)=p\log _{b}x}$	${\textstyle \log _{2}64=\log _{2}\left(2^{6}\right)=6\log _{2}2=6}$
Akar	${\textstyle \log _{b}{\sqrt[{p}]{x}}={\frac {\log _{b}x}{p}}}$	${\textstyle \log _{10}{\sqrt {1000}}={\frac {1}{2}}\log _{10}1000={\frac {3}{2}}=1.5}$

Perubahan basis

Logaritma log_b x dapat dihitung dari logaritma x dan b sehubungan dengan basis arbiter k menggunakan rumus berikut:

$\log _{b}x={\frac {\log _{k}x}{\log _{k}b}}.\,$

Turunan faktor konversi antara logaritma basis arbitrer

Dimulai dari menentukan identitas $x=b^{\log _{b}x}$ kita menerapkan logk pada kedua ruas persamaan ini, untuk mendapatkan $\log _{k}x=\log _{k}\left(b^{\log _{b}x}\right)=\log _{b}x\cdot \log _{k}b$. Memecahkan hasil $\log _{b}x$: $\log _{b}x={\frac {\log _{k}x}{\log _{k}b}}$, menunjukkan faktor konversi dari nilai $\log _{k}$ yang diberikan ke nilai-$\log _{b}$ yang sesuai menjadi $(\log _{k}b)^{-1}.$

Kalkulator ilmiah tipikal menghitung logaritma ke basis 10 dan e. Logaritma sehubungan dengan basis apa pun b ditentukan menggunakan salah satu dari dua logaritma ini dengan rumus sebelumnya:

$\log _{b}x={\frac {\log _{10}x}{\log _{10}b}}={\frac {\log _{e}x}{\log _{e}b}}.$

Diberikan angka x dan logaritmanya y = log_b x ke basis tak-diketahui b, basisnya diberikan oleh:

$b=x^{\frac {1}{y}},$

yang dilihat dari mengambil persamaan pendefinisian $x=b^{\,\log _{b}x}=b^{y}$ ke kuasa (pangkat) ${\tfrac {1}{y}}.$

Basis khusus

Plot logaritma untuk basis 0,5, 2, dan e

Di antara semua pilihan untuk basis, ketiganya adalah basis khusus. Ini adalah b = 10, b = e (irasional adalah konstanta matematika ≈ 2,71828), dan b = 2 (logaritma biner). Dalam analisa matematika, basis logaritma e tersebar luas karena sifat analitik yang dijelaskan dibawah ini. Di sisi lain, logaritma basis-10 mudah digunakan untuk perhitungan manual dalam sistem bilangan desimal:

$\log _{10}(10x)=\log _{10}10+\log _{10}x=1+\log _{10}x.\$

Jadi, log₁₀ (x) berhubungan dengan jumlah digit desimal dari bilangan bulat positif x: jumlah digit adalah bilangan bulat terkecil yang lebih besar dari log₁₀ (x). Misalnya, log₁₀(1430) adalah kira-kira 3,15. Bilangan bulat berikutnya adalah 4, yang merupakan jumlah digit 1430. Baik logaritma alami dan logaritma ke basis dua digunakan dalam teori informasi, sesuai dengan penggunaan nat atau bit sebagai satuan dasar informasi masing-masing. Logaritma biner juga digunakan dalam ilmu komputer, dimana sistem biner ada di mana-mana; dalam teori musik, di mana rasio nada dua (oktaf) ada di mana-mana dan sen adalah logaritma biner (skala 1200) rasio antara dua nada yang bertemperatur sama di musik klasik Eropa; dan dalam fotografi untuk mengukur nilai eksposur.

Tabel berikut mencantumkan notasi umum untuk logaritma ke basis ini dan medan dimana mereka digunakan. Banyak disiplin menulis log x yang dialihkan ke log_b x, ketika dasar yang dimaksud ditentukan dari konteksnya. Notasi ^blog x juga muncul. Kolom "notasi ISO" mencantumkan penunjukan yang disarankan oleh Organisasi Internasional untuk Standardisasi (ISO 80000-2). Karena notasi log x telah digunakan untuk ketiga basis (atau jika basisnya tidak tentu atau tidak material), dasar yang dimaksud seringkali harus disimpulkan berdasarkan konteks atau disiplin. Dalam ilmu komputer, log biasanya mengacu pada log₂, dan dalam matematika log biasanya mengacu pada log_e. Dalam konteks lain, log sering disebut sebagai log₁₀.

Basis b	Nama untuk logb x	Notasi ISO	Notasi lainnya	Digunakan dalam
2	logaritma biner	lb x	ld x, log x, lg x, log2 x	ilmu komputer, teori informasi, bioinformatika, teori musik, fotografi
e	logaritma alami	ln x	log x (dalam matematika dan bahasa pemrograman lainnya), loge x	matematika, fisika, kimia, statistik, ekonomi, teori informasi, dan teknik
10	logaritma umum	lg x	log x, log10 x (dalam bidang teknik, biologi, astronomi)	berbagai bidang teknik (lihat desibel dan lihat di bawah), logaritma tabel, genggam kalkulator, spektroskopi
b	logaritma ke basis b	logb x		matematika

Sejarah

Artikel utama: Sejarah logaritma

Sejarah logaritma dimulai dari Eropa abad ketujuh belas adalah penemuan fungsi baru yang memperluas ranah analisis luar cakupan metode aljabar. Metode logaritma dikemukakan secara terbuka oleh John Napier pada tahun 1614, dalam sebuah buku berjudul Mirifici Logarithmorum Canonis Descriptio (Deskripsi Kaidah Logaritma yang Menakjubkan). Sebelum penemuan Napier, ada teknik lain dengan cakupan serupa, seperti prosthafaeresis atau penggunaan tabel progresi, yang dikembangkan secara ekstensif oleh Jost Bürgi sekitar tahun 1600. Napier menciptakan istilah untuk logaritma dalam bahasa Latin Tengah, “logaritmus” yang berasal dari bahasa Yunani, secara harfiah berarti, “rasio-bilangan,” dari logos “proporsi, rasio, kata” + arithmos “bilangan”.

Logaritma umum suatu bilangan adalah indeks pangkat sepuluh yang sama dengan bilangan tersebut. Berbicara tentang angka yang membutuhkan banyak angka adalah kiasan kasar untuk logaritma umum, dan disebut oleh Archimedes sebagai “urutan bilangan”. Logaritma real pertama adalah metode heuristik untuk mengubah perkalian menjadi penjumlahan, sehingga memudahkan komputasi yang cepat. Beberapa metode ini menggunakan tabel yang diturunkan dari identitas trigonometri. Metode seperti itu disebut prosthafaeresis.

Penemuan fungsi sekarang dikenal sebagai logaritma alami dimulai sebagai upaya untuk kuadratur persegi panjang hiperbola oleh Grégoire de Saint-Vincent ia adalah seorang Yesuit Belgia yang tinggal di Praha. Archimedes telah menulis The Quadrature of the Parabola pada abad ketiga SM, tetapi kuadratur untuk hiperbola menghindari semua upaya sampai Saint-Vincent menerbitkan hasilnya pada tahun 1647. Relasi yang disediakan logaritma antara barisan dan deret geometri dalam argumen dan nilai barisan dan deret aritmetika, A. A. de Sarasa diminta untuk membuat hubungan kuadratur Saint-Vincent dan tradisi logaritma dalam prosthafaeresis, mengarah ke istilah "logaritma hiperbolik", sebuah kata sinonim untuk logaritma alami. Soon the new function was appreciated by Christiaan Huygens, and James Gregory. The notation Log y was adopted by Leibniz in 1675, dan tahun berikutnya dia menghubungkannya ke integral ${\textstyle \int {\frac {dy}{y}}.}$

Sebelum Euler mengembangkan konsep modernnya tentang logaritma alami kompleks, Roger Cotes memiliki hasil yang hampir sama ketika ia menunjukkan pada tahun 1714 bahwa

$\log(\cos \theta +i\sin \theta )=i\theta$.

Tabel logaritma, kaidah geser, dan aplikasi historis

1797 Encyclopædia Britannica penjelasan logaritma

Dengan menyederhanakan perhitungan yang sulit sebelum kalkulator dan komputer tersedia, logaritma berkontribusi pada kemajuan ilmu pengetahuan, khususnya astronomi. Mereka sangat penting untuk kemajuan dalam survei, navigasi langit, dan domain lainnya. Pierre-Simon Laplace menyebutkan tentang logaritma

"...[sebuah] kecerdasan mengagumkan yang mengurangi menjadi beberapa hari kerja berbulan-bulan, menggandakan kehidupan astronom, dan menghindarkannya dari kesalahan dan rasa jijik yang tak terpisahkan dari perhitungan panjang."

Sebagai fungsi $f(x)=b^{x}$ yang merupakan fungsi invers dari $^{b}\!\log x$ disebut juga sebagai antilogaritma. Saat ini, fungsi ini lebih sering disebut fungsi eksponensial.

Tabel log

Alat utama yang memungkinkan penggunaan praktis logaritma adalah tabel logaritma. Tabel tersebut pertama kali disusun oleh Henry Briggs pada tahun 1617, setelah penemuan Napier tetapi dengan inovasi menggunakan 10 sebagai basis. Tabel pertama Briggs berisi logaritma umum dari semua bilangan bulat dalam rentang 1 hingga 1000, dengan presisi 14 digit. Selanjutnya, tabel dengan cakupan yang meningkat ditulis. Tabel ini mencantumkan nilai $^{10}\!\log x$ untuk bilangan apa pun $x$ dalam rentang tertentu, pada presisi tertentu. Logaritma basis-10 digunakan secara universal untuk komputasi, oleh karena itu disebut logaritma umum, karena bilangan yang berbeda dengan faktor 10 memiliki logaritma yang berbeda dengan bilangan bulat. Logaritma umum $x$ dipisahkan menjadi bagian bilangan bulat dan bagian pecahan, yang dikenal sebagai karakteristik dan mantissa. Tabel logaritma hanya perlu menyertakan mantissa, karena karakteristiknya dapat dengan mudah ditentukan dengan menghitung angka dari titik desimal. Karakteristik $10\times x$ adalah satu plus karakteristik $x$, dan mantissanya sama. Jadi dengan menggunakan tabel log tiga digit, logaritma dari 3542 didekati dengan

$^{10}\!\log 3542=^{10}\!\log(1000\cdot 3.542)=3+^{10}\!\log 3.542\approx 3+^{10}\!\log 3.54\,$

Akurasi besar dapat diperoleh dengan interpolasi:

$^{10}\!\log 3542\approx 3+^{10}\!\log 3.54+0.2(^{10}\!\log 3.55-^{10}\!\log 3.54)\,$

Nilai $10^{x}$ dapat ditentukan dengan pencarian terbalik pada tabel yang sama, karena logaritma adalah fungsi monoton.

Komputasi

Hasil kali dan hasil bagi dua bilangan positif $c$ dan $d$ secara rutin dihitung sebagai jumlah dan selisih logaritmanya. Produk $cd$ atau hasil bagi $c/d$ berasal dari antilogaritma dari jumlah atau selisih, melalui tabel yang sama:

$cd=10^{\,^{10}\!\log c}\,10^{\,^{10}\!\log d}=10^{\,^{10}\!\log c\,+\,^{10}\!\log d}$

dan

${\frac {c}{d}}=cd^{-1}=10^{\,^{10}\!\log c\,-\,^{10}\!\log d}.$

Untuk perhitungan manual yang menuntut ketelitian yang cukup besar, melakukan pencarian kedua logaritma, menghitung jumlah atau perbedaannya, dan mencari antilogaritma jauh lebih cepat daripada melakukan perkalian dengan metode sebelumnya seperti prostafaeresis, yang mengandalkan identitas trigonometri.

Perhitungan pangkat dan akar-akar direduksi menjadi perkalian atau pembagian dan pencarian dengan

$c^{d}=\left(10^{\,^{10}\!\log c}\right)^{d}=10^{\,d^{10}\!\log c}$

dan

${\sqrt[{d}]{c}}=c^{\frac {1}{d}}=10^{{\frac {1}{d}}^{10}\!\log c}.$

Perhitungan trigonometri difasilitasi oleh tabel yang berisi logaritma umum fungsi trigonometri.

Kaidah geser

Aplikasi penting lainnya adalah kaidah geser, sepasang skala yang dibagi secara logaritmik yang digunakan dalam perhitungan. Skala logaritmik non-geser, kaidah Gunter, ditemukan tak lama setelah penemuan Napier. William Oughtred menyempurnakannya untuk membuat kaidah geser—sepasang skala logaritmik yang dapat dipindahkan terhadap satu sama lain. Bilangan yang ditempatkan pada skala geser pada jarak sebanding dengan perbedaan antara logaritma mereka. Menggeser skala atas dengan tepat berarti menambahkan logaritma secara mekanis, seperti yang diilustrasikan berikut ini:

Penggambaran skema dari kaidah slide. Mulai dari 2 pada skala bawah, tambahkan jarak ke 3 pada skala atas untuk mencapai produk 6. Kaidah geser berfungsi karena ditandai sedemikian rupa sehingga jarak dari 1 ke $x$ sebanding dengan logaritma $x$.

Misalnya, menambahkan jarak dari 1 ke 2 pada skala yang lebih rendah ke jarak dari 1 ke 3 pada skala atas menghasilkan produk 6, yang dibacakan bagian bawah. Penggaris geser adalah alat penghitung penting bagi para insinyur dan ilmuwan hingga tahun 1970-an, karena memungkinkan, dengan mengorbankan presisi, komputasi yang jauh lebih cepat daripada teknik berdasarkan tabel.

Sifat analitik

Studi logaritma yang lebih mendalam membutuhkan konsep fungsi. Fungsi adalah aturan yang diberikan satu bilangan, yang menghasilkan bilangan lain. Contohnya adalah fungsi yang menghasilkan pangkat $x$ ke-$b$ dari sembarang bilangan real $x$, dengan basis $b$ adalah bilangan tetap. Fungsi ini ditulis sebagai: $f(x)=b^{x}$.

Fungsi logaritma

Untuk membenarkan definisi logaritma, perlu ditunjukkan bahwa persamaan

$b^{x}=y$

memiliki solusi $x$ dan solusi ini unik, asalkan $y$ positif dan $b$ positif dan tidak sama dengan 1. Sebuah bukti dari real itu membutuhkan teorema nilai menengah dari kalkulus dasar. Teorema ini menyatakan bahwa fungsi kontinu yang menghasilkan dua nilai $m$ dan $n$ juga menghasilkan nilai yang terletak antara $m$ dan $n$. Suatu fungsi dikatakan "kontinu" jika tidak "melompat", yaitu jika grafiknya dibuat tanpa pangkat pena.

Sifat ini ditampilkan pada fungsi $f(x)=b^{x}$. Because f takes arbitrarily large and arbitrarily small positive values, any number y > 0 lies between f(x₀) and f(x₁) for suitable x₀ and x₁. Hence, the intermediate value theorem ensures that the equation f(x) = y has a solution. Moreover, there is only one solution to this equation, because the function f is strictly increasing (for b > 1), or strictly decreasing (for 0 < b < 1).

The unique solution x is the logarithm of y to base b, log_b y. The function that assigns to y its logarithm is called logarithm function or logarithmic function (or just logarithm).

The function log_b x is essentially characterized by the product formula

$\log _{b}(xy)=\log _{b}x+\log _{b}y.$

More precisely, the logarithm to any base b > 1 is the only increasing function f from the positive reals to the reals satisfying f(b) = 1 and

$f(xy)=f(x)+f(y).$

Inverse function

The graph of the logarithm function log_b (x) (blue) is obtained by reflecting the graph of the function b^x (red) at the diagonal line (x = y).

The formula for the logarithm of a power says in particular that for any number x,

$\log _{b}\left(b^{x}\right)=x\log _{b}b=x.$

In prose, taking the x-th power of b and then the base-b logarithm gives back x. Conversely, given a positive number y, the formula

$b^{\log _{b}y}=y$

says that first taking the logarithm and then exponentiating gives back y. Thus, the two possible ways of combining (or composing) logarithms and exponentiation give back the original number. Therefore, the logarithm to base b is the inverse function of f(x) = b^x.

Inverse functions are closely related to the original functions. Their graphs correspond to each other upon exchanging the x- and the y-coordinates (or upon reflection at the diagonal line x = y), as shown at the right: a point (t, u = b^t) on the graph of f yields a point (u, t = log_b u) on the graph of the logarithm and vice versa. As a consequence, log_b (x) diverges to infinity (gets bigger than any given number) if x grows to infinity, provided that b is greater than one. In that case, log_b(x) is an increasing function. For b < 1, log_b (x) tends to minus infinity instead. When x approaches zero, log_b x goes to minus infinity for b > 1 (plus infinity for b < 1, respectively).

Derivative and antiderivative

The graph of the natural logarithm (green) and its tangent at x = 1.5 (black)

Analytic properties of functions pass to their inverses. Thus, as f(x) = b^x is a continuous and differentiable function, so is log_b y. Roughly, a continuous function is differentiable if its graph has no sharp "corners". Moreover, as the derivative of f(x) evaluates to ln(b) b^x by the properties of the exponential function, the chain rule implies that the derivative of log_b x is given by

${\frac {d}{dx}}\log _{b}x={\frac {1}{x\ln b}}.$

That is, the slope of the tangent touching the graph of the base-b logarithm at the point (x, log_b (x)) equals 1/(x ln(b)).

The derivative of ln(x) is 1/x; this implies that ln(x) is the unique antiderivative of 1/x that has the value 0 for x = 1. It is this very simple formula that motivated to qualify as "natural" the natural logarithm; this is also one of the main reasons of the importance of the constant e.

The derivative with a generalized functional argument f(x) is

${\frac {d}{dx}}\ln f(x)={\frac {f'(x)}{f(x)}}.$

The quotient at the right hand side is called the logarithmic derivative of f. Computing f'(x) by means of the derivative of ln(f(x)) is known as logarithmic differentiation. The antiderivative of the natural logarithm ln(x) is:

$\int \ln(x)\,dx=x\ln(x)-x+C.$

Related formulas, such as antiderivatives of logarithms to other bases can be derived from this equation using the change of bases.

Integral representation of the natural logarithm

The natural logarithm of t is the shaded area underneath the graph of the function f(x) = 1/x (reciprocal of x).

The natural logarithm of t can be defined as the definite integral:

$\ln t=\int _{1}^{t}{\frac {1}{x}}\,dx.$

This definition has the advantage that it does not rely on the exponential function or any trigonometric functions; the definition is in terms of an integral of a simple reciprocal. As an integral, ln(t) equals the area between the x-axis and the graph of the function 1/x, ranging from x = 1 to x = t. This is a consequence of the fundamental theorem of calculus and the fact that the derivative of ln(x) is 1/x. Product and power logarithm formulas can be derived from this definition. For example, the product formula ln(tu) = ln(t) + ln(u) is deduced as:

$\ln(tu)=\int _{1}^{tu}{\frac {1}{x}}\,dx\ {\stackrel {(1)}{=}}\int _{1}^{t}{\frac {1}{x}}\,dx+\int _{t}^{tu}{\frac {1}{x}}\,dx\ {\stackrel {(2)}{=}}\ln(t)+\int _{1}^{u}{\frac {1}{w}}\,dw=\ln(t)+\ln(u).$

The equality (1) splits the integral into two parts, while the equality (2) is a change of variable (w = x/t). In the illustration below, the splitting corresponds to dividing the area into the yellow and blue parts. Rescaling the left hand blue area vertically by the factor t and shrinking it by the same factor horizontally does not change its size. Moving it appropriately, the area fits the graph of the function f(x) = 1/x again. Therefore, the left hand blue area, which is the integral of f(x) from t to tu is the same as the integral from 1 to u. This justifies the equality (2) with a more geometric proof.

A visual proof of the product formula of the natural logarithm

The power formula ln(t^r) = r ln(t) may be derived in a similar way:

$\ln(t^{r})=\int _{1}^{t^{r}}{\frac {1}{x}}dx=\int _{1}^{t}{\frac {1}{w^{r}}}\left(rw^{r-1}\,dw\right)=r\int _{1}^{t}{\frac {1}{w}}\,dw=r\ln(t).$

The second equality uses a change of variables (integration by substitution), w = x^1/r.

The sum over the reciprocals of natural numbers,

$1+{\frac {1}{2}}+{\frac {1}{3}}+\cdots +{\frac {1}{n}}=\sum _{k=1}^{n}{\frac {1}{k}},$

is called the harmonic series. It is closely tied to the natural logarithm: as n tends to infinity, the difference,

$\sum _{k=1}^{n}{\frac {1}{k}}-\ln(n),$

converges (i.e. gets arbitrarily close) to a number known as the Euler–Mascheroni constant γ = 0.5772.... This relation aids in analyzing the performance of algorithms such as quicksort.

Transcendence of the logarithm

Real numbers that are not algebraic are called transcendental; for example, π and e are such numbers, but ${\sqrt {2-{\sqrt {3}}}}$ is not. Almost all real numbers are transcendental. The logarithm is an example of a transcendental function. The Gelfond–Schneider theorem asserts that logarithms usually take transcendental, i.e. "difficult" values.

Calculation

The logarithm keys (LOG for base 10 and LN for base e) on a TI-83 Plus graphing calculator

Logarithms are easy to compute in some cases, such as log₁₀ (1000) = 3. In general, logarithms can be calculated using power series or the arithmetic–geometric mean, or be retrieved from a precalculated logarithm table that provides a fixed precision. Newton's method, an iterative method to solve equations approximately, can also be used to calculate the logarithm, because its inverse function, the exponential function, can be computed efficiently. Using look-up tables, CORDIC-like methods can be used to compute logarithms by using only the operations of addition and bit shifts. Moreover, the binary logarithm algorithm calculates lb(x) recursively, based on repeated squarings of x, taking advantage of the relation

$\log _{2}\left(x^{2}\right)=2\log _{2}|x|.$

Power series

Taylor series

The Taylor series of ln(z) centered at z = 1. The animation shows the first 10 approximations along with the 99th and 100th. The approximations do not converge beyond a distance of 1 from the center.

For any real number z that satisfies 0 < z ≤ 2, the following formula holds:

${\begin{aligned}\ln(z)&={\frac {(z-1)^{1}}{1}}-{\frac {(z-1)^{2}}{2}}+{\frac {(z-1)^{3}}{3}}-{\frac {(z-1)^{4}}{4}}+\cdots \\&=\sum _{k=1}^{\infty }(-1)^{k+1}{\frac {(z-1)^{k}}{k}}\end{aligned}}$

This is a shorthand for saying that ln(z) can be approximated to a more and more accurate value by the following expressions:

${\begin{array}{lllll}(z-1)&&\\(z-1)&-&{\frac {(z-1)^{2}}{2}}&\\(z-1)&-&{\frac {(z-1)^{2}}{2}}&+&{\frac {(z-1)^{3}}{3}}\\\vdots &\end{array}}$

For example, with z = 1.5 the third approximation yields 0.4167, which is about 0.011 greater than ln(1.5) = 0.405465. This series approximates ln(z) with arbitrary precision, provided the number of summands is large enough. In elementary calculus, ln(z) is therefore the limit of this series. It is the Taylor series of the natural logarithm at z = 1. The Taylor series of ln(z) provides a particularly useful approximation to ln(1 + z) when z is small, |z| < 1, since then

$\ln(1+z)=z-{\frac {z^{2}}{2}}+{\frac {z^{3}}{3}}\cdots \approx z.$

For example, with z = 0.1 the first-order approximation gives ln(1.1) ≈ 0.1, which is less than 5% off the correct value 0.0953.

More efficient series

Another series is based on the area hyperbolic tangent function:

$\ln(z)=2\cdot \operatorname {artanh} \,{\frac {z-1}{z+1}}=2\left({\frac {z-1}{z+1}}+{\frac {1}{3}}{\left({\frac {z-1}{z+1}}\right)}^{3}+{\frac {1}{5}}{\left({\frac {z-1}{z+1}}\right)}^{5}+\cdots \right),$

for any real number z > 0. Using sigma notation, this is also written as

$\ln(z)=2\sum _{k=0}^{\infty }{\frac {1}{2k+1}}\left({\frac {z-1}{z+1}}\right)^{2k+1}.$

This series can be derived from the above Taylor series. It converges more quickly than the Taylor series, especially if z is close to 1. For example, for z = 1.5, the first three terms of the second series approximate ln(1.5) with an error of about 3×10⁻⁶. The quick convergence for z close to 1 can be taken advantage of in the following way: given a low-accuracy approximation y ≈ ln(z) and putting

$A={\frac {z}{\exp(y)}},$

the logarithm of z is:

$\ln(z)=y+\ln(A).$

The better the initial approximation y is, the closer A is to 1, so its logarithm can be calculated efficiently. A can be calculated using the exponential series, which converges quickly provided y is not too large. Calculating the logarithm of larger z can be reduced to smaller values of z by writing z = a · 10^b, so that ln(z) = ln(a) + b · ln(10).

A closely related method can be used to compute the logarithm of integers. Putting $\textstyle z={\frac {n+1}{n}}$ in the above series, it follows that:

$\ln(n+1)=\ln(n)+2\sum _{k=0}^{\infty }{\frac {1}{2k+1}}\left({\frac {1}{2n+1}}\right)^{2k+1}.$

If the logarithm of a large integer n is known, then this series yields a fast converging series for log(n+1), with a rate of convergence of ${\textstyle \left({\frac {1}{2n+1}}\right)^{2}}$.

Arithmetic–geometric mean approximation

The arithmetic–geometric mean yields high precision approximations of the natural logarithm. Sasaki and Kanada showed in 1982 that it was particularly fast for precisions between 400 and 1000 decimal places, while Taylor series methods were typically faster when less precision was needed. In their work ln(x) is approximated to a precision of 2^−p (or p precise bits) by the following formula (due to Carl Friedrich Gauss):

$\ln(x)\approx {\frac {\pi }{2\,\mathrm {M} \!\left(1,2^{2-m}/x\right)}}-m\ln(2).$

Here M(x, y) denotes the arithmetic–geometric mean of x and y. It is obtained by repeatedly calculating the average (x + y)/2 (arithmetic mean) and ${\textstyle {\sqrt {xy}}}$ (geometric mean) of x and y then let those two numbers become the next x and y. The two numbers quickly converge to a common limit which is the value of M(x, y). m is chosen such that

$x\,2^{m}>2^{p/2}.\,$

to ensure the required precision. A larger m makes the M(x, y) calculation take more steps (the initial x and y are farther apart so it takes more steps to converge) but gives more precision. The constants π and ln(2) can be calculated with quickly converging series.

Feynman's algorithm

While at Los Alamos National Laboratory working on the Manhattan Project, Richard Feynman developed a bit-processing algorithm that is similar to long division and was later used in the Connection Machine. The algorithm uses the fact that every real number 1 < x < 2 is representable as a product of distinct factors of the form 1 + 2^−k. The algorithm sequentially builds that product P: if P · (1 + 2^−k) < x, then it changes P to P · (1 + 2^−k). It then increases $k$ by one regardless. The algorithm stops when k is large enough to give the desired accuracy. Because log(x) is the sum of the terms of the form log(1 + 2^−k) corresponding to those k for which the factor 1 + 2^−k was included in the product P, log(x) may be computed by simple addition, using a table of log(1 + 2^−k) for all k. Any base may be used for the logarithm table.

Applications

A nautilus displaying a logarithmic spiral

Logarithms have many applications inside and outside mathematics. Some of these occurrences are related to the notion of scale invariance. For example, each chamber of the shell of a nautilus is an approximate copy of the next one, scaled by a constant factor. This gives rise to a logarithmic spiral. Benford's law on the distribution of leading digits can also be explained by scale invariance. Logarithms are also linked to self-similarity. For example, logarithms appear in the analysis of algorithms that solve a problem by dividing it into two similar smaller problems and patching their solutions. The dimensions of self-similar geometric shapes, that is, shapes whose parts resemble the overall picture are also based on logarithms. Logarithmic scales are useful for quantifying the relative change of a value as opposed to its absolute difference. Moreover, because the logarithmic function log(x) grows very slowly for large x, logarithmic scales are used to compress large-scale scientific data. Logarithms also occur in numerous scientific formulas, such as the Tsiolkovsky rocket equation, the Fenske equation, or the Nernst equation.

Logarithmic scale

Artikel utama: Logarithmic scale

A logarithmic chart depicting the value of one Goldmark in Papiermarks during the German hyperinflation in the 1920s

Scientific quantities are often expressed as logarithms of other quantities, using a logarithmic scale. For example, the decibel is a unit of measurement associated with logarithmic-scale quantities. It is based on the common logarithm of ratios—10 times the common logarithm of a power ratio or 20 times the common logarithm of a voltage ratio. It is used to quantify the loss of voltage levels in transmitting electrical signals, to describe power levels of sounds in acoustics, and the absorbance of light in the fields of spectrometry and optics. The signal-to-noise ratio describing the amount of unwanted noise in relation to a (meaningful) signal is also measured in decibels. In a similar vein, the peak signal-to-noise ratio is commonly used to assess the quality of sound and image compression methods using the logarithm.

The strength of an earthquake is measured by taking the common logarithm of the energy emitted at the quake. This is used in the moment magnitude scale or the Richter magnitude scale. For example, a 5.0 earthquake releases 32 times (10^1.5) and a 6.0 releases 1000 times (10³) the energy of a 4.0. Apparent magnitude measures the brightness of stars logarithmically. In chemistry the negative of the decimal logarithm, the decimal cologarithm, is indicated by the letter p. For instance, pH is the decimal cologarithm of the activity of hydronium ions (the form hydrogen ions H⁺ take in water). The activity of hydronium ions in neutral water is 10⁻⁷ mol·L⁻¹, hence a pH of 7. Vinegar typically has a pH of about 3. The difference of 4 corresponds to a ratio of 10⁴ of the activity, that is, vinegar's hydronium ion activity is about 10⁻³ mol·L⁻¹.

Semilog (log–linear) graphs use the logarithmic scale concept for visualization: one axis, typically the vertical one, is scaled logarithmically. For example, the chart at the right compresses the steep increase from 1 million to 1 trillion to the same space (on the vertical axis) as the increase from 1 to 1 million. In such graphs, exponential functions of the form f(x) = a · b^x appear as straight lines with slope equal to the logarithm of b. Log-log graphs scale both axes logarithmically, which causes functions of the form f(x) = a · x^k to be depicted as straight lines with slope equal to the exponent k. This is applied in visualizing and analyzing power laws.

Psychology

Logarithms occur in several laws describing human perception: Hick's law proposes a logarithmic relation between the time individuals take to choose an alternative and the number of choices they have. Fitts's law predicts that the time required to rapidly move to a target area is a logarithmic function of the distance to and the size of the target. In psychophysics, the Weber–Fechner law proposes a logarithmic relationship between stimulus and sensation such as the actual vs. the perceived weight of an item a person is carrying. (This "law", however, is less realistic than more recent models, such as Stevens's power law.)

Psychological studies found that individuals with little mathematics education tend to estimate quantities logarithmically, that is, they position a number on an unmarked line according to its logarithm, so that 10 is positioned as close to 100 as 100 is to 1000. Increasing education shifts this to a linear estimate (positioning 1000 10 times as far away) in some circumstances, while logarithms are used when the numbers to be plotted are difficult to plot linearly.

Probability theory and statistics

Three probability density functions (PDF) of random variables with log-normal distributions. The location parameter μ, which is zero for all three of the PDFs shown, is the mean of the logarithm of the random variable, not the mean of the variable itself.

Distribution of first digits (in %, red bars) in the population of the 237 countries of the world. Black dots indicate the distribution predicted by Benford's law.

Logarithms arise in probability theory: the law of large numbers dictates that, for a fair coin, as the number of coin-tosses increases to infinity, the observed proportion of heads approaches one-half. The fluctuations of this proportion about one-half are described by the law of the iterated logarithm.

Logarithms also occur in log-normal distributions. When the logarithm of a random variable has a normal distribution, the variable is said to have a log-normal distribution. Log-normal distributions are encountered in many fields, wherever a variable is formed as the product of many independent positive random variables, for example in the study of turbulence.

Logarithms are used for maximum-likelihood estimation of parametric statistical models. For such a model, the likelihood function depends on at least one parameter that must be estimated. A maximum of the likelihood function occurs at the same parameter-value as a maximum of the logarithm of the likelihood (the "log likelihood"), because the logarithm is an increasing function. The log-likelihood is easier to maximize, especially for the multiplied likelihoods for independent random variables.

Benford's law describes the occurrence of digits in many data sets, such as heights of buildings. According to Benford's law, the probability that the first decimal-digit of an item in the data sample is d (from 1 to 9) equals log₁₀ (d + 1) − log₁₀ (d), regardless of the unit of measurement. Thus, about 30% of the data can be expected to have 1 as first digit, 18% start with 2, etc. Auditors examine deviations from Benford's law to detect fraudulent accounting.

Computational complexity

Analysis of algorithms is a branch of computer science that studies the performance of algorithms (computer programs solving a certain problem). Logarithms are valuable for describing algorithms that divide a problem into smaller ones, and join the solutions of the subproblems.

For example, to find a number in a sorted list, the binary search algorithm checks the middle entry and proceeds with the half before or after the middle entry if the number is still not found. This algorithm requires, on average, log₂ (N) comparisons, where N is the list's length. Similarly, the merge sort algorithm sorts an unsorted list by dividing the list into halves and sorting these first before merging the results. Merge sort algorithms typically require a time approximately proportional to N · log(N). The base of the logarithm is not specified here, because the result only changes by a constant factor when another base is used. A constant factor is usually disregarded in the analysis of algorithms under the standard uniform cost model.

A function f(x) is said to grow logarithmically if f(x) is (exactly or approximately) proportional to the logarithm of x. (Biological descriptions of organism growth, however, use this term for an exponential function.) For example, any natural number N can be represented in binary form in no more than log₂ N + 1 bits. In other words, the amount of memory needed to store N grows logarithmically with N.

Entropy and chaos

Billiards on an oval billiard table. Two particles, starting at the center with an angle differing by one degree, take paths that diverge chaotically because of reflections at the boundary.

Entropy is broadly a measure of the disorder of some system. In statistical thermodynamics, the entropy S of some physical system is defined as

$S=-k\sum _{i}p_{i}\ln(p_{i}).\,$

The sum is over all possible states i of the system in question, such as the positions of gas particles in a container. Moreover, p_i is the probability that the state i is attained and k is the Boltzmann constant. Similarly, entropy in information theory measures the quantity of information. If a message recipient may expect any one of N possible messages with equal likelihood, then the amount of information conveyed by any one such message is quantified as log₂ N bits.

Lyapunov exponents use logarithms to gauge the degree of chaoticity of a dynamical system. For example, for a particle moving on an oval billiard table, even small changes of the initial conditions result in very different paths of the particle. Such systems are chaotic in a deterministic way, because small measurement errors of the initial state predictably lead to largely different final states. At least one Lyapunov exponent of a deterministically chaotic system is positive.

Fractals

The Sierpinski triangle (at the right) is constructed by repeatedly replacing equilateral triangles by three smaller ones.

Logarithms occur in definitions of the dimension of fractals. Fractals are geometric objects that are self-similar: small parts reproduce, at least roughly, the entire global structure. The Sierpinski triangle (pictured) can be covered by three copies of itself, each having sides half the original length. This makes the Hausdorff dimension of this structure ln(3)/ln(2) ≈ 1.58. Another logarithm-based notion of dimension is obtained by counting the number of boxes needed to cover the fractal in question.

Music

Four different octaves shown on a linear scale, then shown on a logarithmic scale (as the ear hears them).

Logarithms are related to musical tones and intervals. In equal temperament, the frequency ratio depends only on the interval between two tones, not on the specific frequency, or pitch, of the individual tones. For example, the note A has a frequency of 440 Hz and B-flat has a frequency of 466 Hz. The interval between A and B-flat is a semitone, as is the one between B-flat and B (frequency 493 Hz). Accordingly, the frequency ratios agree:

${\frac {466}{440}}\approx {\frac {493}{466}}\approx 1.059\approx {\sqrt[{12}]{2}}.$

Therefore, logarithms can be used to describe the intervals: an interval is measured in semitones by taking the base-2^1/12 logarithm of the frequency ratio, while the base-2^1/1200 logarithm of the frequency ratio expresses the interval in cents, hundredths of a semitone. The latter is used for finer encoding, as it is needed for non-equal temperaments.

Interval (the two tones are played at the same time)	1/12 tone playⓘ	Semitone playⓘ	Just major third playⓘ	Major third playⓘ	Tritone playⓘ	Octave playⓘ
Frequency ratio r	$2^{\frac {1}{72}}\approx 1.0097$	$2^{\frac {1}{12}}\approx 1.0595$	${\tfrac {5}{4}}=1.25$	${\begin{aligned}2^{\frac {4}{12}}&={\sqrt[{3}]{2}}\\&\approx 1.2599\end{aligned}}$	${\begin{aligned}2^{\frac {6}{12}}&={\sqrt {2}}\\&\approx 1.4142\end{aligned}}$	$2^{\frac {12}{12}}=2$
Corresponding number of semitones $\log _{\sqrt[{12}]{2}}(r)=12\log _{2}(r)$	${\tfrac {1}{6}}$	$1$	$\approx 3.8631$	$4$	$6$	$12$
Corresponding number of cents $\log _{\sqrt[{1200}]{2}}(r)=1200\log _{2}(r)$	$16{\tfrac {2}{3}}$	$100$	$\approx 386.31$	$400$	$600$	$1200$

Number theory

Natural logarithms are closely linked to counting prime numbers (2, 3, 5, 7, 11, ...), an important topic in number theory. For any integer x, the quantity of prime numbers less than or equal to x is denoted π(x). The prime number theorem asserts that π(x) is approximately given by

${\frac {x}{\ln(x)}},$

in the sense that the ratio of π(x) and that fraction approaches 1 when x tends to infinity. As a consequence, the probability that a randomly chosen number between 1 and x is prime is inversely proportional to the number of decimal digits of x. A far better estimate of π(x) is given by the offset logarithmic integral function Li(x), defined by

$\mathrm {Li} (x)=\int _{2}^{x}{\frac {1}{\ln(t)}}\,dt.$

The Riemann hypothesis, one of the oldest open mathematical conjectures, can be stated in terms of comparing π(x) and Li(x). The Erdős–Kac theorem describing the number of distinct prime factors also involves the natural logarithm.

The logarithm of n factorial, n! = 1 · 2 · ... · n, is given by

$\ln(n!)=\ln(1)+\ln(2)+\cdots +\ln(n).$

This can be used to obtain Stirling's formula, an approximation of n! for large n.

Generalizations

Complex logarithm

Artikel utama: Complex logarithm

Polar form of z = x + iy. Both φ and φ' are arguments of z.

All the complex numbers a that solve the equation

$e^{a}=z$

are called complex logarithms of z, when z is (considered as) a complex number. A complex number is commonly represented as z = x + iy, where x and y are real numbers and i is an imaginary unit, the square of which is −1. Such a number can be visualized by a point in the complex plane, as shown at the right. The polar form encodes a non-zero complex number z by its absolute value, that is, the (positive, real) distance r to the origin, and an angle between the real (x) axis Re and the line passing through both the origin and z. This angle is called the argument of z.

The absolute value r of z is given by

$\textstyle r={\sqrt {x^{2}+y^{2}}}.$

Using the geometrical interpretation of sine and cosine and their periodicity in 2π, any complex number z may be denoted as

$z=x+iy=r(\cos \varphi +i\sin \varphi )=r(\cos(\varphi +2k\pi )+i\sin(\varphi +2k\pi )),$

for any integer number k. Evidently the argument of z is not uniquely specified: both φ and φ' = φ + 2kπ are valid arguments of z for all integers k, because adding 2kπ radians or k⋅360° to φ corresponds to "winding" around the origin counter-clock-wise by k turns. The resulting complex number is always z, as illustrated at the right for k = 1. One may select exactly one of the possible arguments of z as the so-called principal argument, denoted Arg(z), with a capital A, by requiring φ to belong to one, conveniently selected turn, e.g. −π < φ ≤ π or 0 ≤ φ < 2π. These regions, where the argument of z is uniquely determined are called branches of the argument function.

The principal branch (-π, π) of the complex logarithm, Log(z). The black point at z = 1 corresponds to absolute value zero and brighter, more saturated colors refer to bigger absolute values. The hue of the color encodes the argument of Log(z).

Euler's formula connects the trigonometric functions sine and cosine to the complex exponential:

$e^{i\varphi }=\cos \varphi +i\sin \varphi .$

Using this formula, and again the periodicity, the following identities hold:

${\begin{array}{lll}z&=&r\left(\cos \varphi +i\sin \varphi \right)\\&=&r\left(\cos(\varphi +2k\pi )+i\sin(\varphi +2k\pi )\right)\\&=&re^{i(\varphi +2k\pi )}\\&=&e^{\ln(r)}e^{i(\varphi +2k\pi )}\\&=&e^{\ln(r)+i(\varphi +2k\pi )}=e^{a_{k}},\end{array}}$

where ln(r) is the unique real natural logarithm, a_k denote the complex logarithms of z, and k is an arbitrary integer. Therefore, the complex logarithms of z, which are all those complex values a_k for which the a_k-th power of e equals z, are the infinitely many values

$a_{k}=\ln(r)+i(\varphi +2k\pi ),\quad$ for arbitrary integers k.

Taking k such that φ + 2kπ is within the defined interval for the principal arguments, then a_k is called the principal value of the logarithm, denoted Log(z), again with a capital L. The principal argument of any positive real number x is 0; hence Log(x) is a real number and equals the real (natural) logarithm. However, the above formulas for logarithms of products and powers do not generalize to the principal value of the complex logarithm.

The illustration at the right depicts Log(z), confining the arguments of z to the interval (−π, π]. This way the corresponding branch of the complex logarithm has discontinuities all along the negative real x axis, which can be seen in the jump in the hue there. This discontinuity arises from jumping to the other boundary in the same branch, when crossing a boundary, i.e. not changing to the corresponding k-value of the continuously neighboring branch. Such a locus is called a branch cut. Dropping the range restrictions on the argument makes the relations "argument of z", and consequently the "logarithm of z", multi-valued functions.

Inverses of other exponential functions

Exponentiation occurs in many areas of mathematics and its inverse function is often referred to as the logarithm. For example, the logarithm of a matrix is the (multi-valued) inverse function of the matrix exponential. Another example is the p-adic logarithm, the inverse function of the p-adic exponential. Both are defined via Taylor series analogous to the real case. In the context of differential geometry, the exponential map maps the tangent space at a point of a manifold to a neighborhood of that point. Its inverse is also called the logarithmic (or log) map.

In the context of finite groups exponentiation is given by repeatedly multiplying one group element b with itself. The discrete logarithm is the integer n solving the equation

$b^{n}=x,$

where x is an element of the group. Carrying out the exponentiation can be done efficiently, but the discrete logarithm is believed to be very hard to calculate in some groups. This asymmetry has important applications in public key cryptography, such as for example in the Diffie–Hellman key exchange, a routine that allows secure exchanges of cryptographic keys over unsecured information channels. Zech's logarithm is related to the discrete logarithm in the multiplicative group of non-zero elements of a finite field.

Further logarithm-like inverse functions include the double logarithm ln(ln(x)), the super- or hyper-4-logarithm (a slight variation of which is called iterated logarithm in computer science), the Lambert W function, and the logit. They are the inverse functions of the double exponential function, tetration, of f(w) = we^w, and of the logistic function, respectively.

Related concepts

From the perspective of group theory, the identity log(cd) = log(c) + log(d) expresses a group isomorphism between positive reals under multiplication and reals under addition. Logarithmic functions are the only continuous isomorphisms between these groups. By means of that isomorphism, the Haar measure (Lebesgue measure) dx on the reals corresponds to the Haar measure dx/x on the positive reals. The non-negative reals not only have a multiplication, but also have addition, and form a semiring, called the probability semiring; this is in fact a semifield. The logarithm then takes multiplication to addition (log multiplication), and takes addition to log addition (LogSumExp), giving an isomorphism of semirings between the probability semiring and the log semiring.

Logarithmic one-forms df/f appear in complex analysis and algebraic geometry as differential forms with logarithmic poles.

The polylogarithm is the function defined by

$\operatorname {Li} _{s}(z)=\sum _{k=1}^{\infty }{z^{k} \over k^{s}}.$

It is related to the natural logarithm by Li₁ (z) = −ln(1 − z). Moreover, Li_s (1) equals the Riemann zeta function ζ(s).

See also

• Cologarithm
• Decimal exponent (dex)
• Exponential function
• Index of logarithm articles
• Logarithmic notation

External links

• Media tentang Klasüo/bak pasir di Wikimedia Commons
• Definisi kamus klasüo/bak pasir di Wikikamus
• (Inggris) Weisstein, Eric W., "Logarithm", MathWorld 
• Khan Academy: Logarithms, free online micro lectures
• Hazewinkel, Michiel, ed. (2001) [1994], "Logarithmic function", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4 
• Colin Byfleet, Educational video on logarithms, diakses tanggal 12 October 2010 
• Edward Wright, Translation of Napier's work on logarithms, diarsipkan dari versi asli tanggal 3 December 2002, diakses tanggal 12 October 2010 
•  Glaisher, James Whitbread Lee (1911), "Logarithm", dalam Chisholm, Hugh, Encyclopædia Britannica, 16 (edisi ke-11), Cambridge University Press, hlm. 868–77