\documentclass[12pt,titlepage]{article}

\usepackage{amsmath}
\usepackage{mathrsfs}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{mathtools}
\usepackage{graphicx}
\usepackage{color}
\usepackage{ucs}
\usepackage[utf8x]{inputenc}
\usepackage{xparse}
\usepackage{hyperref}

%----Macros----------
%
% Unresolved issues:
%
%  \righttoleftarrow
%  \lefttorightarrow
%
%  \color{} with HTML colorspec
%  \bgcolor
%  \array with options (without options, it's equivalent to the matrix environment)

% Of the standard HTML named colors, white, black, red, green, blue and yellow
% are predefined in the color package. Here are the rest.
\definecolor{aqua}{rgb}{0, 1.0, 1.0}
\definecolor{fuschia}{rgb}{1.0, 0, 1.0}
\definecolor{gray}{rgb}{0.502, 0.502, 0.502}
\definecolor{lime}{rgb}{0, 1.0, 0}
\definecolor{maroon}{rgb}{0.502, 0, 0}
\definecolor{navy}{rgb}{0, 0, 0.502}
\definecolor{olive}{rgb}{0.502, 0.502, 0}
\definecolor{purple}{rgb}{0.502, 0, 0.502}
\definecolor{silver}{rgb}{0.753, 0.753, 0.753}
\definecolor{teal}{rgb}{0, 0.502, 0.502}

% Because of conflicts, \space and \mathop are converted to
% \itexspace and \operatorname during preprocessing.

% itex: \space{ht}{dp}{wd}
%
% Height and baseline depth measurements are in units of tenths of an ex while
% the width is measured in tenths of an em.
\makeatletter
\newdimen\itex@wd%
\newdimen\itex@dp%
\newdimen\itex@thd%
\def\itexspace#1#2#3{\itex@wd=#3em%
\itex@wd=0.1\itex@wd%
\itex@dp=#2ex%
\itex@dp=0.1\itex@dp%
\itex@thd=#1ex%
\itex@thd=0.1\itex@thd%
\advance\itex@thd\the\itex@dp%
\makebox[\the\itex@wd]{\rule[-\the\itex@dp]{0cm}{\the\itex@thd}}}
\makeatother

% \tensor and \multiscript
\makeatletter
\newif\if@sup
\newtoks\@sups
\def\append@sup#1{\edef\act{\noexpand\@sups={\the\@sups #1}}\act}%
\def\reset@sup{\@supfalse\@sups={}}%
\def\mk@scripts#1#2{\if #2/ \if@sup ^{\the\@sups}\fi \else%
  \ifx #1_ \if@sup ^{\the\@sups}\reset@sup \fi {}_{#2}%
  \else \append@sup#2 \@suptrue \fi%
  \expandafter\mk@scripts\fi}
\def\tensor#1#2{\reset@sup#1\mk@scripts#2_/}
\def\multiscripts#1#2#3{\reset@sup{}\mk@scripts#1_/#2%
  \reset@sup\mk@scripts#3_/}
\makeatother

% \slash
\makeatletter
\newbox\slashbox \setbox\slashbox=\hbox{$/$}
\def\itex@pslash#1{\setbox\@tempboxa=\hbox{$#1$}
  \@tempdima=0.5\wd\slashbox \advance\@tempdima 0.5\wd\@tempboxa
  \copy\slashbox \kern-\@tempdima \box\@tempboxa}
\def\slash{\protect\itex@pslash}
\makeatother

% math-mode versions of \rlap, etc
% from Alexander Perlis, "A complement to \smash, \llap, and lap"
%   http://math.arizona.edu/~aprl/publications/mathclap/
\def\clap#1{\hbox to 0pt{\hss#1\hss}}
\def\mathllap{\mathpalette\mathllapinternal}
\def\mathrlap{\mathpalette\mathrlapinternal}
\def\mathclap{\mathpalette\mathclapinternal}
\def\mathllapinternal#1#2{\llap{$\mathsurround=0pt#1{#2}$}}
\def\mathrlapinternal#1#2{\rlap{$\mathsurround=0pt#1{#2}$}}
\def\mathclapinternal#1#2{\clap{$\mathsurround=0pt#1{#2}$}}

% Renames \sqrt as \oldsqrt and redefine root to result in \sqrt[#1]{#2}
\let\oldroot\root
\def\root#1#2{\oldroot #1 \of{#2}}
\renewcommand{\sqrt}[2][]{\oldroot #1 \of{#2}}

% Manually declare the txfonts symbolsC font
\DeclareSymbolFont{symbolsC}{U}{txsyc}{m}{n}
\SetSymbolFont{symbolsC}{bold}{U}{txsyc}{bx}{n}
\DeclareFontSubstitution{U}{txsyc}{m}{n}

% Manually declare the stmaryrd font
\DeclareSymbolFont{stmry}{U}{stmry}{m}{n}
\SetSymbolFont{stmry}{bold}{U}{stmry}{b}{n}

% Manually declare the MnSymbolE font
\DeclareFontFamily{OMX}{MnSymbolE}{}
\DeclareSymbolFont{mnomx}{OMX}{MnSymbolE}{m}{n}
\SetSymbolFont{mnomx}{bold}{OMX}{MnSymbolE}{b}{n}
\DeclareFontShape{OMX}{MnSymbolE}{m}{n}{
    <-6>  MnSymbolE5
   <6-7>  MnSymbolE6
   <7-8>  MnSymbolE7
   <8-9>  MnSymbolE8
   <9-10> MnSymbolE9
  <10-12> MnSymbolE10
  <12->   MnSymbolE12}{}

% Declare specific arrows from txfonts without loading the full package
\makeatletter
\def\re@DeclareMathSymbol#1#2#3#4{%
    \let#1=\undefined
    \DeclareMathSymbol{#1}{#2}{#3}{#4}}
\re@DeclareMathSymbol{\neArrow}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\neArr}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\seArrow}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\seArr}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\nwArrow}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\nwArr}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\swArrow}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\swArr}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\nequiv}{\mathrel}{symbolsC}{46}
\re@DeclareMathSymbol{\Perp}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\Vbar}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\sslash}{\mathrel}{stmry}{12}
\re@DeclareMathSymbol{\bigsqcap}{\mathop}{stmry}{"64}
\re@DeclareMathSymbol{\biginterleave}{\mathop}{stmry}{"6}
\re@DeclareMathSymbol{\invamp}{\mathrel}{symbolsC}{77}
\re@DeclareMathSymbol{\parr}{\mathrel}{symbolsC}{77}
\makeatother

% \llangle, \rrangle, \lmoustache and \rmoustache from MnSymbolE
\makeatletter
\def\Decl@Mn@Delim#1#2#3#4{%
  \if\relax\noexpand#1%
    \let#1\undefined
  \fi
  \DeclareMathDelimiter{#1}{#2}{#3}{#4}{#3}{#4}}
\def\Decl@Mn@Open#1#2#3{\Decl@Mn@Delim{#1}{\mathopen}{#2}{#3}}
\def\Decl@Mn@Close#1#2#3{\Decl@Mn@Delim{#1}{\mathclose}{#2}{#3}}
\Decl@Mn@Open{\llangle}{mnomx}{'164}
\Decl@Mn@Close{\rrangle}{mnomx}{'171}
\Decl@Mn@Open{\lmoustache}{mnomx}{'245}
\Decl@Mn@Close{\rmoustache}{mnomx}{'244}
\makeatother

% Widecheck
\makeatletter
\DeclareRobustCommand\widecheck[1]{{\mathpalette\@widecheck{#1}}}
\def\@widecheck#1#2{%
    \setbox\z@\hbox{\m@th$#1#2$}%
    \setbox\tw@\hbox{\m@th$#1%
       \widehat{%
          \vrule\@width\z@\@height\ht\z@
          \vrule\@height\z@\@width\wd\z@}$}%
    \dp\tw@-\ht\z@
    \@tempdima\ht\z@ \advance\@tempdima2\ht\tw@ \divide\@tempdima\thr@@
    \setbox\tw@\hbox{%
       \raise\@tempdima\hbox{\scalebox{1}[-1]{\lower\@tempdima\box
\tw@}}}%
    {\ooalign{\box\tw@ \cr \box\z@}}}
\makeatother

% \mathraisebox{voffset}[height][depth]{something}
\makeatletter
\NewDocumentCommand\mathraisebox{moom}{%
\IfNoValueTF{#2}{\def\@temp##1##2{\raisebox{#1}{$\m@th##1##2$}}}{%
\IfNoValueTF{#3}{\def\@temp##1##2{\raisebox{#1}[#2]{$\m@th##1##2$}}%
}{\def\@temp##1##2{\raisebox{#1}[#2][#3]{$\m@th##1##2$}}}}%
\mathpalette\@temp{#4}}
\makeatletter

% udots (taken from yhmath)
\makeatletter
\def\udots{\mathinner{\mkern2mu\raise\p@\hbox{.}
\mkern2mu\raise4\p@\hbox{.}\mkern1mu
\raise7\p@\vbox{\kern7\p@\hbox{.}}\mkern1mu}}
\makeatother

%% Fix array
\newcommand{\itexarray}[1]{\begin{matrix}#1\end{matrix}}
%% \itexnum is a noop
\newcommand{\itexnum}[1]{#1}

%% Renaming existing commands
\newcommand{\underoverset}[3]{\underset{#1}{\overset{#2}{#3}}}
\newcommand{\widevec}{\overrightarrow}
\newcommand{\darr}{\downarrow}
\newcommand{\nearr}{\nearrow}
\newcommand{\nwarr}{\nwarrow}
\newcommand{\searr}{\searrow}
\newcommand{\swarr}{\swarrow}
\newcommand{\curvearrowbotright}{\curvearrowright}
\newcommand{\uparr}{\uparrow}
\newcommand{\downuparrow}{\updownarrow}
\newcommand{\duparr}{\updownarrow}
\newcommand{\updarr}{\updownarrow}
\newcommand{\gt}{>}
\newcommand{\lt}{<}
\newcommand{\map}{\mapsto}
\newcommand{\embedsin}{\hookrightarrow}
\newcommand{\Alpha}{A}
\newcommand{\Beta}{B}
\newcommand{\Zeta}{Z}
\newcommand{\Eta}{H}
\newcommand{\Iota}{I}
\newcommand{\Kappa}{K}
\newcommand{\Mu}{M}
\newcommand{\Nu}{N}
\newcommand{\Rho}{P}
\newcommand{\Tau}{T}
\newcommand{\Upsi}{\Upsilon}
\newcommand{\omicron}{o}
\newcommand{\lang}{\langle}
\newcommand{\rang}{\rangle}
\newcommand{\Union}{\bigcup}
\newcommand{\Intersection}{\bigcap}
\newcommand{\Oplus}{\bigoplus}
\newcommand{\Otimes}{\bigotimes}
\newcommand{\Wedge}{\bigwedge}
\newcommand{\Vee}{\bigvee}
\newcommand{\coproduct}{\coprod}
\newcommand{\product}{\prod}
\newcommand{\closure}{\overline}
\newcommand{\integral}{\int}
\newcommand{\doubleintegral}{\iint}
\newcommand{\tripleintegral}{\iiint}
\newcommand{\quadrupleintegral}{\iiiint}
\newcommand{\conint}{\oint}
\newcommand{\contourintegral}{\oint}
\newcommand{\infinity}{\infty}
\newcommand{\bottom}{\bot}
\newcommand{\minusb}{\boxminus}
\newcommand{\plusb}{\boxplus}
\newcommand{\timesb}{\boxtimes}
\newcommand{\intersection}{\cap}
\newcommand{\union}{\cup}
\newcommand{\Del}{\nabla}
\newcommand{\odash}{\circleddash}
\newcommand{\negspace}{\!}
\newcommand{\widebar}{\overline}
\newcommand{\textsize}{\normalsize}
\renewcommand{\scriptsize}{\scriptstyle}
\newcommand{\scriptscriptsize}{\scriptscriptstyle}
\newcommand{\mathfr}{\mathfrak}
\newcommand{\statusline}[2]{#2}
\newcommand{\tooltip}[2]{#2}
\newcommand{\toggle}[2]{#2}

% Theorem Environments
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{prop}{Proposition}
\newtheorem{cor}{Corollary}
\newtheorem*{utheorem}{Theorem}
\newtheorem*{ulemma}{Lemma}
\newtheorem*{uprop}{Proposition}
\newtheorem*{ucor}{Corollary}
\theoremstyle{definition}
\newtheorem{defn}{Definition}
\newtheorem{example}{Example}
\newtheorem*{udefn}{Definition}
\newtheorem*{uexample}{Example}
\theoremstyle{remark}
\newtheorem{remark}{Remark}
\newtheorem{note}{Note}
\newtheorem*{uremark}{Remark}
\newtheorem*{unote}{Note}

%-------------------------------------------------------------------

\begin{document}

%-------------------------------------------------------------------


\section*{logarithm}

\hypertarget{logarithms}{}\section*{{Logarithms}}\label{logarithms}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{logarithms_of_real_numbers}{Logarithms of real numbers}\dotfill \pageref*{logarithms_of_real_numbers} \linebreak
\noindent\hyperlink{logarithms_of_complex_numbers}{Logarithms of complex numbers}\dotfill \pageref*{logarithms_of_complex_numbers} \linebreak
\noindent\hyperlink{logarithms_and_lie_groups}{Logarithms and Lie groups}\dotfill \pageref*{logarithms_and_lie_groups} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

Classically, a logarithm is a [[partial function|partially-defined]] [[smooth map|smooth]] [[homomorphism]] from a multiplicative [[group]] of [[number]]s to an additive group of numbers. As such, it is a [[local section]] of an [[exponential map]]. As exponential maps can be generalised to [[Lie groups]], so can logarithms.

\hypertarget{definitions}{}\subsection*{{Definitions}}\label{definitions}

\hypertarget{logarithms_of_real_numbers}{}\subsubsection*{{Logarithms of real numbers}}\label{logarithms_of_real_numbers}

Consider the [[field]] of [[real numbers]]; these numbers form a [[Lie group]] under addition (which we will call simply $\mathbb{R}$), while the nonzero numbers form a Lie group under multiplication (which we will call $\mathbb{R}^*$). The multiplicative group has two [[connected components]]; we will focus attention on the [[identity component]] (which we will call $\mathbb{R}^+$), consisting of the positive numbers.

The Lie groups $\mathbb{R}$ and $\mathbb{R}^+$ are in fact [[isomorphic]]. In fact, there is one isomorphism for each positive real number $b$ other than $1$; this number $b$ is called the \textbf{base}. Fixing a base, the map from $\mathbb{R}^+$ to $\mathbb{R}$ is called the \textbf{real logarithm with base $b$}, written $x \mapsto \log_b x$; the map from $\mathbb{R}$ to $\mathbb{R}^+$ is the \textbf{real [[exponential map]] with base $b$}, written $x \mapsto b^x$.

The real logarithms are handily defined using the [[Riemann integral]] as follows:

\begin{equation}
\array {
   \ln x & \coloneqq \int_1^x \frac{1}{t} \,\mathrm{d}t ;\\
   \log_b x & \coloneqq \frac{\ln x}{\ln b} .\\
}
\label{integrals}\end{equation}
Note that $\ln$ is itself a logarithm, the \textbf{natural logarithm}, whose base is $\mathrm{e} = 2.71828182845\ldots$. (The exponential map may similarly be defined as an infinite series, but I'll leave that for its own article.)

\hypertarget{logarithms_of_complex_numbers}{}\subsubsection*{{Logarithms of complex numbers}}\label{logarithms_of_complex_numbers}

Now consider the [[field]] of [[complex numbers]]; these also form a [[Lie group]] under addition (which we call $\mathbb{C}$), while the nonzero numbers form a Lie group under multiplication (which we call $\mathbb{C}^*$). Now the multiplicative group is [[connected space|connected]], so we would like to use all of it.

However, $\mathbb{C}$ and $\mathbb{C}^*$ are \emph{not} [[isomorphic]]. Indeed, the multiplication map

\begin{displaymath}
\mathbb{R}^* \times S^1 \to \mathbb{C}^*
\end{displaymath}
exhibits $\mathbb{C}^*$ as a [[biproduct]] of $\mathbb{R}^*$ and the [[circle group]] $S^1$, so that homomorphisms $\mathbb{C}^* \to \mathbb{C}$ are given by pairs of homomorphisms $f \colon \mathbb{R}^* \to \mathbb{C}$, $g \colon S^1 \to \mathbb{C}$. But every homomorphisms $g \colon S^1 \to \mathbb{C}$ is trivial: the restriction of $g$ to the [[torsion subgroup]] of $S^1$ is trivial since $\mathbb{C}$ is torsionfree, and since the torsion subgroup is dense in $S^1$, any Lie group homomorphism $S^1 \to \mathbb{C}$ must also be trivial. Therefore, every homomorphism $h \colon \mathbb{C}^* \to \mathbb{C}$ factors through the projection $\mathbb{C}^* \to \mathbb{R}^*$. It quickly follows that no such $h$ can be injective, nor can such $h$ be surjective.

Taking advantage of biproduct representations $\mathbb{C} \cong \mathbb{R} \oplus \mathbb{R}$ and $\mathbb{C}^* \cong \mathbb{R}^* \oplus S^1$, we can classify homomorphisms from $\mathbb{C}$ to $\mathbb{C}^*$. Each is given by a 4-tuple of real numbers $(a, b, c, d)$:

\begin{displaymath}
\phi_{a, b, c, d}(x + i y) = e^{a x} e^{i b x} e^{c y} e^{i d y}.
\end{displaymath}
The cases where $a = d$, $b = -c$ correspond to those homomorphisms that are [[holomorphic functions]] (i.e., that satisfy the [[Cauchy-Riemann equations]]). Putting $w = a + b i$, we have

\begin{displaymath}
\phi_{a, b, -b, a}(z) = e^{w z}
\end{displaymath}
with one such homomorphism for each complex number $w$, and these homomorphisms are [[surjections]] whenever $w \ne 0$. (N.B.: these homomorphisms are not uniquely determined by their values at $z = 1$, since we have $e^w = e^{w'}$ whenever $w - w'$ is an integer multiple of $2 \pi i$, and yet the homomorphisms $z \mapsto e^{w z}$ and $z \mapsto e^{w' z}$ will be different unless $w = w'$.)

So we have these surjections (the \textbf{complex [[exponential map]]} $z \mapsto e^{w z}$, for $w \ne 0$), which are [[regular epimorphisms]] but not [[split epimorphisms]]. However, while they have no [[sections]] (being not split), they have quite a few [[local sections]], and the [[domains]] of the [[maximal local sections]] are precisely the [[connected space|connected]] [[simply connected space|simply connected]] [[open subspace|open]] [[dense subspace|dense]] subspaces $R$ of $\mathbb{C}^*$. A \textbf{complex logarithm with exponential base $w$ on $R$} is this $R$-defined section of the complex exponential map $z \mapsto e^{w z}$. Supposing $R$ given, we denote this by $\log_{[w]}$ (but please note that in the context of real logarithms, this would ordinarily be denoted $\log_b$ where $b = e^w$).

If $1 \in R$, then a complex natural logarithm on $R$ may be defined using the [[contour integral]] with the same formula \eqref{integrals} as for the real natural logarithm. We merely insist that the integral be done along a contour within the region $R$. (Since $R$ is connected, there is such a contour; since $R$ is simply connected and $t \mapsto 1/t$ is [[holomorphic map|holomorphic]], the result is unique.) Note that if $x \in \mathbb{R}^+ \subseteq R$, then the real and complex natural logarithms of $x$ will be equal.

The natural exponential map is [[periodic function|periodic]] (with period $2 \pi \mathrm{i}$), and it is possible to add any multiple of this period to the natural logarithm of any $x \ne 1$ by suitably changing the region $R$. We then obtain the most general notion of maximally-defined complex logarithm with any base by using the formulas

\begin{displaymath}
\array {
   \ln x & \coloneqq C + \int_{\mathrm{e}^C}^x \frac{1}{t} \,\mathrm{d}t,\\
   \log_{[w]} x & \coloneqq \frac{\ln x}{w} .\\
}
\end{displaymath}
\hypertarget{logarithms_and_lie_groups}{}\subsubsection*{{Logarithms and Lie groups}}\label{logarithms_and_lie_groups}

In the classical examples, the multiplicative groups $\mathbb{R}^+$ and $\mathbb{C}^*$ both [[Lie groups]]. The additive groups $\mathbb{R}$ and $\mathbb{C}$ are also Lie groups, but they are more than this: they are [[Lie algebras]]. (The additive group of a Lie algebra is always a Lie group. Actually, since these are [[abelian Lie algebras]], their Lie-algebra structure is easy to miss, but of course they are [[vector spaces]].) And what's more, each additive group is \emph{the} Lie algebra of the corresponding Lie group.

This generalises. Given any [[Lie group]] $G$, let $\mathfrak{g}$ be its [[Lie algebra]]. Then we have an [[exponential map]] $\exp\colon \mathfrak{g} \to G$, which is [[surjection|surjective]] under certain conditions (most famously when $G$ is [[connected space|connected]] and [[compact space|compact]], but also in the classical cases, even though $G$ is not compact). More generally, given any [[automorphism]] $\phi$ of $\mathfrak{g}$, we have a map $x \mapsto \exp(\phi(x))$, which is a [[homomorphism]] of Lie groups. Any [[local section]] of this map may be called a \textbf{logarithm base $\phi$} on $G$ (denoted $\log_{[\phi]}$ with the bracket as in the previous section); any local section of $\exp$ itself may be called a \textbf{natural logarithm} on $G$.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[logarithmic integral function]]


\item [[logarithmic cohomology operation]]



\end{itemize}
[[!redirects logarithm]]

[[!redirects logarithms]]

[[!redirects logarithmic]]

[[!redirects natural logarithm]]

[[!redirects natural logarithms]]



\end{document}