Pengguna:Dedhert.Jr/Uji halaman 01/16: Perbedaan antara revisi

Logaritma dengan bilangan pokok {{mvar|b}} merupakan operasi invers yang menyediakan nilai keluar {{mvar|y}} dari nilai masukan {{mvar|x}}. Dalam artian, <math>y = \, ^b\!\log x</math> ekuivalen dengan to <math>x=b^y</math>, jika {{mvar|b}} [[bilangan real]] positif. (Jika {{mvar|b}} bukanlah bilangan real positif, eksponensiasi dan logaritma dapat didefinisikan, namun memberikan beberapa nilai, sehingga definisi darinya semakin rumit.)

: <math>^b\!\log(xy)= \,^b\!\log x + \,^b\!\log y,</math>

Rumus tersebut dapat diperlihatkan dengan mengambil persamaan yang mendefinisikan <math> x = b^{^b\!\logblog x} = b^y</math>, lalu dipangkatkan dengan <math>\tfrac{1}{y}.</math>

Tabel berikut memuat notasi-notasi umum mengenai bilangan pokok beserta bidang yang dipakai. Ada beberapa mata pelajaran yang menulis {{math|log&thinsp;''x''}} daripada {{math|log_''b''&thinsp;''x''}}, dan adapula notasi {{math|^''b''log&thinsp;''x''}} yang juga muncul pada beberapa mata pelajaran.<ref>{{Citation|url=http://www.mathe-online.at/mathint/lexikon/l.html|author1=Franz Embacher|author2=Petra Oberhuemer|title=Mathematisches Lexikon|publisher=mathe online: für Schule, Fachhochschule, Universität unde Selbststudium|access-date=22 March 2011|language=de}}</ref> Pada kolom "Notasi ISO" memuat penamaan yang disarankan berdasarkanoleh [[InternationalOrganisasi OrganizationStandardisasi for StandardizationInternasional]], yakni ([[ISO 80000-2]]).<ref>Quantities and units – Part 2: Mathematics (ISO 80000-2:2019); EN ISO 80000-2</ref> Karena notasi {{math|log {{mvar|x}}}} telah dipakai untuk ketiga bilangan pokok di atas (atau ketika bilangan pokok belum ditentukan), bilangan pokok yang dimaksud harus sering diduga tergantung konteks atau mata pelajarannya. Sebagai contoh, {{Math|log}} biasanya mengacu pada {{math|²log}} dalam ilmu komputer, dan {{Math|log}} mengacu pada {{math|^''e''log}}.<ref>{{citation|first1=Michael T.|last1=Goodrich|author1-link=Michael T. Goodrich|first2=Roberto|last2=Tamassia|author2-link=Roberto Tamassia|title=Algorithm Design: Foundations, Analysis, and Internet Examples|publisher=John Wiley & Sons|date=2002|page=23|quote=One of the interesting and sometimes even surprising aspects of the analysis of data structures and algorithms is the ubiquitous presence of logarithms&nbsp;... As is the custom in the computing literature, we omit writing the base {{mvar|b}} of the logarithm when {{math|1=''b'' = 2}}.}}</ref> Dalam konteks lainnya, {{Math|log}} seringkali mengacu pada {{math|log<subsup>10</subsup>log}}.<ref>{{citation|title=Introduction to Applied Mathematics for Environmental Science|edition=illustrated|first1=David F.|last1=Parkhurst|publisher=Springer Science & Business Media|date=2007|isbn=978-0-387-34228-3|page=288|url={{google books |plainurl=y |id=h6yq_lOr8Z4C|page=288 }}}}</ref>

|{{math|log ''x''}}, {{math|log<subsup>10</subsup>log&thinsp;''x''}}

[[Berkas:Logarithms_Britannica_1797.png|ka|jmpl|360x360px|Penjelasan logaritma didalam ''[[Encyclopædia Britannica]]'' pada tahun 1797.]]

Karena fungsi {{math|''f''(''x'') {{=}} {{mvar|b}}^''x''}} merupakan fungsi invers dari {{math|^''b''log&thinsp;''x''}}, maka fungsi tersebut disebut sebagai '''antilogaritma'''.<ref>{{Citation|editor1-last=Abramowitz|editor1-first=Milton|editor1-link=Milton Abramowitz|editor2-last=Stegun|editor2-first=Irene A.|editor2-link=Irene Stegun|title=Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables|publisher=[[Dover Publications]]|location=New York|isbn=978-0-486-61272-0|edition=10th|year=1972|title-link=Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables}}, section 4.7., p.&nbsp;89</ref> Saat ini, fungsi tersebutantilogaritma lebih sering disebut [[fungsi eksponensial]].

ASebuah keyalat toolpenting thatyang enabledmemungkinkan thepenggunaan practicallogaritma useadalah of logarithms was the ''[[Log table|table oftabel logarithmslogaritma]]''.<ref>{{Citation|last1=Campbell-Kelly|first1=Martin|title=The history of mathematical tables: from Sumer to spreadsheets|title-link=The History of Mathematical Tables|publisher=[[Oxford University Press]]|series=Oxford scholarship online|isbn=978-0-19-850841-0|year=2003}}, section 2</ref> TheTabel firstpertama suchdisusun table was compiled byoleh [[Henry Briggs (mathematician)|Henry Briggs]] inpada tahun 1617, immediatelysetelah afterpenemuan Napier's, inventionnamun butpenemuannya with the innovation of usingmenggunakan 10 assebagai thebilangan basepokok. Briggs' first tableTabel containedpertamanya thememuat [[Commonlogaritma logarithm|common logarithmsbiasa]] ofdari allsemua integersbilangan inbulat theyang rangeberkisar fromantara 1 todengan 1000, withyang amemiliki precisionketepatan of14 14&nbsp;digitsdigit. Subsequently, tables with increasing scope were written. These tables listed the values of {{math|log₁₀&thinsp;''x''}} for any number&nbsp;{{mvar|x}} in a certain range, at a certain precision. Base-10 logarithms were universally used for computation, hence the name common logarithm, since numbers that differ by factors of 10 have logarithms that differ by integers. The common logarithm of {{mvar|x}} can be separated into an [[integer part]] and a [[fractional part]], known as the characteristic and [[Mantissa (logarithm)|mantissa]]. Tables of logarithms need only include the mantissa, as the characteristic can be easily determined by counting digits from the decimal point.<ref>{{Citation|last1=Spiegel|first1=Murray R.|last2=Moyer|first2=R.E.|title=Schaum's outline of college algebra|publisher=[[McGraw-Hill]]|location=New York|series=Schaum's outline series|isbn=978-0-07-145227-4|year=2006}}, p.&nbsp;264</ref> The characteristic of {{math|10 · {{mvar|x}}}} is one plus the characteristic of {{mvar|x}}, and their mantissas are the same. Thus using a three-digit log table, the logarithm of 3542 is approximated by