In mathematics, there are several logarithmic identities.

Algebraic identities

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Using simpler operations

People use logarithms to make calculations easier. For instance, two numbers can be multiplied just by using a logarithm table and adding.

$\backslash log\_b(xy)\; =\; \backslash log\_b(x)\; +\; \backslash log\_b(y)\; \backslash !\backslash ,$	because	$b^x\; \backslash cdot\; b^y\; =\; b^\{x\; +\; y\}$
$\backslash log\_b\backslash !\backslash left(\backslash begin\{matrix\}\backslash frac\{x\}\{y\}\backslash end\{matrix\}\backslash right)\; =\; \backslash log\_b(x)\; -\; \backslash log\_b(y)$	because	$\backslash begin\{matrix\}\backslash frac\{b^x\}\{b^y\}\backslash end\{matrix\}\; =\; b^\{x\; -\; y\}$
$\backslash log\_b(x^y)\; =\; y\; \backslash log\_b(x)\; \backslash !\backslash ,$	because	$(b^x)^y\; =\; b^\{xy\}\; \backslash !\backslash ,$
$\backslash log\_b\backslash !\backslash left(\backslash !\backslash sqrt=\; \backslash begin\{matrix\}\backslash frac\{\backslash log\_b(x)\}\{y\}\backslash end\{matrix\}$	because	$\backslash sqrt[y\{x\}\; =\; x^\{1/y\}$

Cancelling exponentials

Logarithms and exponentials (antilogarithms) with the same base cancel each other.

$b^\{\backslash log\_b(x)\}\; =\; x$	because	$\backslash mathrm\{antilog\}\_b(\backslash log\_b(x))\; =\; x\; \backslash !\backslash ,$
$\backslash log\_b(b^x)\; =\; x\; \backslash !\backslash ,$	because	$\backslash log\_b(\backslash mathrm\{antilog\}\_b(x))\; =\; x\; \backslash !\backslash ,$

Keep in mind that these are true only for real x; the logarithm is infinite-valued on the complex plane, and thus the complex exponential is not invertable.

Changing the base

$\backslash log\_a\; b\; =\; \{\backslash log\_c\; b\; \backslash over\; \backslash log\_c\; a\}$

This identity is needed to evaluate logarithms on calculators. For instance, most calculators have buttons for ln and for log₁₀, but not for log₂. To find log₂(3), you have to calculate log₁₀(3) / log₁₀(2) (or ln(3)/ln(2), which is the same thing).

This formula has several consequences:

$\backslash log\_a\; b\; =\; \backslash frac\{1\}\{\backslash log\_b\; a\}$

$\backslash log\_\{a^n\}\; b\; =\; \backslash frac\{1\}\{n\}\; \backslash log\_a\; b$

$a^\{\backslash log\_b\; c\}\; =\; c^\{\backslash log\_b\; a\}$

Logarithmic "snake" identity

Any sequence

$\backslash !\backslash \; \backslash log\_a(b)\; \backslash cdot\; \backslash log\_b(c)\; \backslash cdot\; \backslash log\_c(d)\backslash cdots\; \backslash log\_y(z)$

can be simplified to...

$\backslash !\backslash \; \backslash log\_a(z)$

This leads to the identity

$\backslash log\_a\; b\; =\; \{\backslash log\_c\; b\; \backslash over\; \backslash log\_c\; a\}.\backslash ,$

Trivial identities

$\backslash log\_b(1)\; =\; 0\; \backslash !\backslash ,$	because	$b^0\; =\; 1\backslash !\backslash ,$
$\backslash log\_b(b)\; =\; 1\; \backslash !\backslash ,$	because	$b^1\; =\; b\backslash !\backslash ,$

$\backslash log\_b(0)$ is undefined because there is no number $x$ such that $b^x\; =\; 0$.

Calculus identities

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Limits

$\backslash lim\_\{x\; \backslash to\; 0^+\}\; \backslash log\_a\; x\; =\; -\backslash infty\; \backslash quad\; \backslash mbox\{if\; \}\; a\; >\; 1$

$\backslash lim\_\{x\; \backslash to\; \backslash infty\}\; \backslash log\_a\; x\; =\; \backslash infty\; \backslash quad\; \backslash mbox\{if\; \}\; a\; >\; 1$

$\backslash lim\_\{x\; \backslash to\; 0^+\}\; x^b\; \backslash log\_a\; x\; =\; 0$

$\backslash lim\_\{x\; \backslash to\; \backslash infty\}\; \{1\; \backslash over\; x^b\}\; \backslash log\_a\; x\; =\; 0$

The last limit is often summarized as "logarithms grow more slowly than any power or root of x".

note: to say the limit of a function "equals infinity" is not strictly correct notation, as "infinity" is not a value. What is meant by the limits equations above is simply that the functions increase/decrease without bound.

Derivatives of logarithmic functions

$\{d\; \backslash over\; dx\}\; \backslash log\_a\; x\; =\; \{1\; \backslash over\; x\; \backslash ln\; a\}\; =\; \{\backslash log\_a\; e\; \backslash over\; x\; \}$

Integral definition

$\backslash log\_e\; x\; =\; \backslash int\_1^x\; \backslash frac\; \{1\}\{t\}\; dt$

Integrals of logarithmic functions

$\backslash int\; \backslash log\_a\; x\; \backslash ,\; dx\; =\; x(\backslash log\_a\; x\; -\; \backslash log\_a\; e)\; +\; C$

To remember higher integrals, it's convenient to define:

$x^\{\backslash leftn\; \backslash right\}\; =\; x^\{n\}(\backslash log(x)\; -\; H\_n)$

where $H\_n$ is the nth harmonic number. So, for example, the first few are:

$x^\{\backslash left0\; \backslash right\}\; =\; \backslash log\; x$

$x^\{\backslash left1\; \backslash right\}\; =\; x\; \backslash log(x)\; -\; x$

$x^\{\backslash left2\; \backslash right\}\; =\; x^2\; \backslash log(x)\; -\; \backslash begin\{matrix\}\; \backslash frac\{3\}\{2\}\; \backslash end\{matrix\}\; \backslash ,\; x^2$

$x^\{\backslash left3\; \backslash right\}\; =\; x^3\; \backslash log(x)\; -\; \backslash begin\{matrix\}\; \backslash frac\{11\}\{6\}\; \backslash end\{matrix\}\; \backslash ,\; x^3$

Then,

$\backslash frac\; \{d\}\{dx\}\; \backslash ,\; x^\{\backslash leftn\; \backslash right\}\; =\; n\; \backslash ,\; x^\{\backslash leftn-1\; \backslash right\}$

$\backslash int\; x^\{\backslash leftn\; \backslash right\}\backslash ,dx\; =\; \backslash frac\; \{x^\{\backslash leftn+1\; \backslash right\}\}\; \{n+1\}\; +\; C$

Logarithms | Mathematical identities

Identidades logarítmicas | identités logarithmiques | Identità sui logaritmi | логаритамске једначине