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%%% Primary Title: logarithm
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%%% Filename: Logarithm.tex
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%%% Owner: rmilson
%%% Author(s): kfgauss70, Mathprof, Wkbj79, rmilson
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\begin{document}

 \paragraph{Definition.}
Three \htmladdnormallink{real numbers}{http://planetmath.org/encyclopedia/MathbbR.html} $x, y, p$, with $x,y>0$ and $x \neq 1$, are said to obey the
logarithmic \htmladdnormallink{relation}{http://planetmath.org/encyclopedia/Surjective2.html}
$$\log_x(y)= p$$
if they obey the corresponding \htmladdnormallink{exponential}{http://planetmath.org/encyclopedia/TrigonometricFunction.html} relation:
$$x^p=y.$$
Note that by the \htmladdnormallink{monotonicity}{http://planetmath.org/encyclopedia/OrderHomomorphism.html} and continuity \htmladdnormallink{property of the
exponential}{http://planetmath.org/encyclopedia/PropertiesOfTheExponential.html} \htmladdnormallink{operation}{http://planetmath.org/encyclopedia/Operation.html}, for given $x$ and $y$ there exists a unique $p$
satisfying the above relation. We are therefore able to says that
{\em $p$ is the logarithm of $y$ relative to the base $x$.}

\paragraph{Properties.}
There are a \htmladdnormallink{number}{http://planetmath.org/encyclopedia/Number.html} of basic \htmladdnormallink{algebraic}{http://planetmath.org/encyclopedia/Transcendental.html} \htmladdnormallink{identities}{http://planetmath.org/encyclopedia/Bijective3.html} involving logarithms.
\begin{align*}
\log_x(yz) &= \log_x(y) + \log_x(z)\\
\log_x \left( \frac{y}{z} \right) &= \log_x(y) - \log_x(z)\\
\log_x(y^z) &= z \log_x(y) \\
\log_x(1) &= 0 \\
\log_x(x) &= 1 \\
\log_x(y) \log_y(x) &= 1\\
\log_y(z) &= \frac{\log_x(z)}{\log_x(y)}
\end{align*}

By the very first identity, any logarithm \htmladdnormallink{restricted}{http://planetmath.org/encyclopedia/RestrictionOfAFunction.html} to the set of \htmladdnormallink{positive}{http://planetmath.org/encyclopedia/Negative.html} \htmladdnormallink{integers}{http://planetmath.org/encyclopedia/RationalInteger.html} is an \htmladdnormallink{additive function}{http://planetmath.org/encyclopedia/AdditiveFunction2.html}.

{\bf Notes.} In essence, logarithms convert \htmladdnormallink{multiplication}{http://planetmath.org/encyclopedia/Multiplication.html} to
\htmladdnormallink{addition}{http://planetmath.org/encyclopedia/Addition.html}, and exponentiation to multiplication. Historically, these
\htmladdnormallink{properties}{http://planetmath.org/encyclopedia/Predicate.html} of the logarithm made it a useful tool for doing numerical
calculations. Before the advent of electronic \htmladdnormallink{calculators}{http://planetmath.org/encyclopedia/Calculator.html} and
computers, tables of logarithms and the logarithmic slide rule were
essential computational aids.

Scientific \htmladdnormallink{applications}{http://planetmath.org/encyclopedia/Applications.html} predominantly make use of
logarithms whose base is the \htmladdnormallink{Eulerian number}{http://planetmath.org/encyclopedia/NapiersConstant.html} $e = 2.71828\ldots$.
Such logarithms are called {\em natural logarithms} and are commonly
denoted by the symbol $\ln$, e.g.
$$\ln(e) = 1.$$
Natural logarithms naturally give rise to the natural logarithm \htmladdnormallink{function}{http://planetmath.org/encyclopedia/Surjective2.html}.

A frequent convention, seen in elementary mathematics texts and on
calculators, is that logarithms that do not give a base explicitly are
assumed to be base $10$, e.g.
$$\log(100) = 2.$$
This is far from universal. In Rudin's ``Real and
Complex analysis'', for example, we see a baseless $\log$ used to
refer to the natural logarithm. By contrast, computer science and
\htmladdnormallink{information}{http://planetmath.org/encyclopedia/CramerRaoLowerBound.html} \htmladdnormallink{theory}{http://planetmath.org/encyclopedia/FinitelyAxiomatizableTheory.html} texts often assume 2 as the default logarithm base.
This is motivated by the fact that $\log_2(N)$ is the approximate
number of bits required to encode $N$ different messages.


The invention of logarithms is commonly credited to John Napier [
\htmladdnormallink{Biography}{http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Napier.html}]

\end{document}