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: <math>\textstyle r=\sqrt{x^2+y^2}.</math>
Dengan menggunakan pandangan geometris fungsi [[sinus (matematikatrigonometri)|sinus]] dan [[kosinus (matematikatrigonometri)|kosinus]] beserta periodisitasnya dalam {{Math|2{{pi}}}}, setiap bilangan kompleks {{mvar|z}} dapat dinyatakan sebagai
: <math>z = x + iy = r (\cos \varphi + i \sin \varphi )= r (\cos (\varphi + 2k\pi) + i \sin (\varphi + 2k\pi)),</math>
untuk setiap bilangan bulat {{mvar|k}}. EvidentlyNyatanya theargumen argument ofdari {{mvar|z}} istidak notdijelaskan uniquelysecara specifiedunik: bothbilangan {{mvar|φ}} anddan {{Math|1=''φ''' = ''φ'' + 2''k''{{pi}}}} aremerupakan argumen valid arguments ofdari {{mvar|z}} foruntuk semua allbilangan integersbulat {{mvar|k}}, becausekarena addingmenambahkan {{Math|2''k''{{pi}}}} [[Radian|radiansradian]] oratau ''k''⋅360°{{refn|SeeLihat [[radian]] foruntuk thekonversi conversion betweenantara 2[[pi|{{pi}}]] anddengan 360 [[degreederajat (anglesudut)|degreederajat]].|group=nb}} toke bilangan {{mvar|φ}} corresponds to "winding" around the origin counter-clock-wise by {{mvar|k}} [[Turn (geometry)|turns]]. The resulting complex number is always {{mvar|z}}, as illustrated at the right for {{math|''k'' {{=}} 1}}. One may select exactly one of the possible arguments of {{mvar|z}} as the so-called ''principal argument'', denoted {{math|Arg(''z'')}}, with a capital {{math|A}}, by requiring {{mvar|φ}} to belong to one, conveniently selected turn, e.g. {{Math|−{{pi}} < ''φ'' ≤ {{pi}}}}<ref>{{Citation|last1=Ganguly|location=Kolkata|first1=S.|title=Elements of Complex Analysis|publisher=Academic Publishers|isbn=978-81-87504-86-3|year=2005}}, Definition 1.6.3</ref> or {{Math|0 ≤ ''φ'' < 2{{pi}}}}.<ref>{{Citation|last1=Nevanlinna|first1=Rolf Herman|author1-link=Rolf Nevanlinna|last2=Paatero|first2=Veikko|title=Introduction to complex analysis|journal=London: Hilger|location=Providence, RI|publisher=AMS Bookstore|isbn=978-0-8218-4399-4|year=2007|bibcode=1974aitc.book.....W}}, section 5.9</ref> These regions, where the argument of {{mvar|z}} is uniquely determined are called [[Principal branch|''branches'']] of the argument function.
[[Berkas:Complex_log_domain.svg|al=A density plot. In the middle there is a black point, at the negative axis the hue jumps sharply and evolves smoothly otherwise.|ka|jmpl|The principal branch (-{{pi}}, {{pi}}) of the complex logarithm, {{math|Log(''z'')}}. The black point at {{math|''z'' {{=}} 1}} corresponds to absolute value zero and brighter colors refer to bigger absolute values. The [[hue]] of the color encodes the argument of {{math|Log(''z'')}}.]]
[[Rumus Euler]] mengaitkan [[fungsi trigonometri]] [[sinus (trigonometri|sinus]] dan [[kosinus (trigonometri)|kosinus]] dengan [[Rumus Euler|eksponensial kompleks]]:
[[Euler's formula]] connects the [[trigonometric functions]] [[sine]] and [[cosine]] to the [[complex exponential]]:
: <math>e^{i\varphi} = \cos \varphi + i\sin \varphi .</math>
UsingDengan thismenggunakan formula,rumus anddi atas, againdan theperiodisitasnya periodicitylagi, themaka followingberlaku identitiesidentitas holdberikut:<ref>{{Citation|last1=Moore|first1=Theral Orvis|last2=Hadlock|first2=Edwin H.|title=Complex analysis|publisher=[[World Scientific]]|location=Singapore|isbn=978-981-02-0246-0|year=1991}}, section 1.2</ref>
: <math> \begin{array}{lll}z& = & r \left (\cos \varphi + i \sin \varphi\right) \\
</math>
wheredengan {{math|ln(''r'')}} isadalah thefungsi uniquelogaritma real natural logarithmtunggal, {{math|''a''<sub>''k''</sub>}} denotemenyatakan thelogaritma complexkompleks logarithms ofdari {{mvar|z}}, anddan {{mvar|k}} isbilangan anbulat arbitrary integersembarang. ThereforeKarena itu, the complexlogaritma logarithmskompleks ofdari {{mvar|z}}, whichyang aresemua allbilangan those complex valueskompleks {{math|''a''<sub>''k''</sub>}} foruntuk which{{mvar|e}} thepangkat {{math|''a''<sub>''k''</sub>-th}} power of {{mvar|e}}sama equalsdengan {{mvar|z}}, aremempunyai thetak infinitelyberhingga manybanyaknya valuesnilai
: <math>a_k = \ln (r) + i ( \varphi + 2 k \pi ),\quad</math> foruntuk arbitrarybilangan integersbulat semabrang {{mvar|k}}.
Taking {{mvar|k}} such that {{Math|''φ'' + 2''k''{{pi}}}} is within the defined interval for the principal arguments, then {{math|''a''<sub>''k''</sub>}} is called the ''principal value'' of the logarithm, denoted {{math|Log(''z'')}}, again with a capital {{math|L}}. The principal argument of any positive real number {{mvar|x}} is 0; hence {{math|Log(''x'')}} is a real number and equals the real (natural) logarithm. However, the above formulas for logarithms of products and powers [[Exponentiation#Failure of power and logarithm identities|do ''not'' generalize]] to the principal value of the complex logarithm.<ref>{{Citation|last1=Wilde|first1=Ivan Francis|title=Lecture notes on complex analysis|publisher=Imperial College Press|location=London|isbn=978-1-86094-642-4|year=2006|url=https://books.google.com/books?id=vrWES2W6vG0C&q=complex+logarithm&pg=PA97}}, theorem 6.1.</ref>
The illustration at the right depicts {{math|Log(''z'')}}, confining the arguments of {{mvar|z}} to the interval {{open-closed|−π, π}}. This way the corresponding branch of the complex logarithm has discontinuities all along the negative real {{mvar|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 {{mvar|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 {{mvar|z}}", and consequently the "logarithm of {{mvar|z}}", [[Multi-valued function|multi-valued functions]].
=== Kebalikan dari fungsi eksponensial lainnya ===
ExponentiationEksponensiasi occursmuncul indalam manycabang areasmatematika ofdan mathematicsfungsi andinversnya itsseringkali inversemengacu functionpada is often referred to as the logarithmlogaritma. ForSebagai examplecontoh, the [[logarithm of alogaritma matrixmatriks]] ismerupakan thefungsi invers (multi-valuedbernilai banyak) inverse function of thedari [[matrixeksponensial exponentialmatriks]].<ref>{{Citation|last1=Higham|first1=Nicholas|author1-link=Nicholas Higham|title=Functions of Matrices. Theory and Computation|location=Philadelphia, PA|publisher=[[Society for Industrial and Applied Mathematics|SIAM]]|isbn=978-0-89871-646-7|year=2008}}, chapter 11.</ref> AnotherContohnya examplelain is theseperti [[PFungsi logaritma p-adicadik|fungsi logarithmlogaritma function|''p''-adicadik logarithm]], thefungsi inverseinvers functiondari of[[fungsi theeksponensial [[Pp-adicadik|fungsi exponentialeksponensial function|''p''-adic exponentialadik]]. BothKedua arefungsi definedtersebut viadidefinisikan melalui deret Taylor seriesyang analogousanalog todengan thekasus realbilangan casereal.<ref>{{Neukirch ANT|mode=cs2}}, section II.5.</ref> InDalam the context ofkonteks [[differentialgeometri geometrydiferensial]], the [[ExponentialPeta mapeksponensial (Riemanniangeometri geometryRiemann)|exponentialpeta mapeksponensial]] maps thememetakan [[tangentruang garis spacesinggung]] atdi asebuah point of atitik [[DifferentiableManifold manifoldterdiferensialkan|manifold]] to ake [[NeighborhoodTetangga (mathematicsmatematika)|neighborhoodtetangga]] oftitik tersebutof that point. Its inverse is also called the logarithmic (or log) map.<ref>{{Citation|last1=Hancock|first1=Edwin R.|last2=Martin|first2=Ralph R.|last3=Sabin|first3=Malcolm A.|title=Mathematics of Surfaces XIII: 13th IMA International Conference York, UK, September 7–9, 2009 Proceedings|url=https://books.google.com/books?id=0cqCy9x7V_QC&pg=PA379|publisher=Springer|year=2009|page=379|isbn=978-3-642-03595-1}}</ref>
In the context of [[finite groups]] exponentiation is given by repeatedly multiplying one group element {{mvar|b}} with itself. The [[discrete logarithm]] is the integer ''{{mvar|n}}'' solving the equation
=== Konsep yang berkaitan ===
FromBerdasarkan thesudut perspective ofpandang [[groupteori theorygrup]], the identityidentitas {{math|log(''cd'') {{=}} log(''c'') + log(''d'')}} expresses amenyatakan [[groupisomorfisme isomorphismgrup]] betweenantara positivebilangan [[Realbilangan numberreal|realsreal]] underpositif multiplicationterhadap andperkalian realsbilangan underreal addition.positif Logarithmicterhadap functions arepenambahan. theFungsi onlylogaritmik continuoushanya isomorphismsisomorfisme betweenkontinu theseantara groupsgrup.<ref>{{Citation|last1=Bourbaki|first1=Nicolas|author1-link=Nicolas Bourbaki|title=General topology. Chapters 5–10|publisher=[[Springer-Verlag]]|location=Berlin, New York|series=Elements of Mathematics|isbn=978-3-540-64563-4|mr=1726872|year=1998}}, section V.4.1</ref> ByBerdasarkan meanspengertian ofisomorfisme that isomorphismtersebut, the [[ukuran Haar measure]] ([[Lebesgueukuran measureLebesgue]]) {{math|''dx''}} onpada thereal realsberpadanan correspondsdengan to theukuran Haar measure {{math|{{sfrac|1=''dx''/|2=''x''}}}} onpada thebilangan positivereal realspositif.<ref>{{Citation|last1=Ambartzumian|first1=R.V.|author-link=Rouben V. Ambartzumian|title=Factorization calculus and geometric probability|publisher=[[Cambridge University Press]]|isbn=978-0-521-34535-4|year=1990|url-access=registration|url=https://archive.org/details/factorizationcal0000amba}}, section 1.4</ref> TheBilangan non-negativereal realstaknegatif nottidak onlyhanya haveterhadap aoperasi multiplicationperkalian, butnamun alsojuga haveterhadap additionoperasi penambahan, anddan formbilangan areal taknegatif membentuk [[semiringsemigelanggang]], calledyang thedisebut sebagai [[probabilitysemigelanggang semiringprobabilitas]]; 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 form|Logarithmic one-forms ]]{{math|''df''/''f''}} appear in [[complex analysis]] and [[algebraic geometry]] as [[Differential form|differential forms]] with logarithmic [[Pole (complex analysis)|poles]].<ref>{{Citation|last1=Esnault|first1=Hélène|last2=Viehweg|first2=Eckart|title=Lectures on vanishing theorems|location=Basel, Boston|publisher=Birkhäuser Verlag|series=DMV Seminar|isbn=978-3-7643-2822-1|mr=1193913|year=1992|volume=20|doi=10.1007/978-3-0348-8600-0|citeseerx=10.1.1.178.3227}}, section 2</ref>
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