# Logarithm

## Definition of logarithm

In mathematics, logarithm of a given number x is the exponent to which another fixed number, the base b, must be raised, to produce that number x:

\begin{align} \mathrm{log}_{a}b&=x \Leftrightarrow b=a^{x},\; \textrm{where}\; a > 0\; \textrm{ja}\; a\neq 1\\ \end{align}

Logarithm is the inverse function to exponentiation.

The logarithm base 10 (that is b = 10) is called the decimal or common logarithm.

$$\mathrm{log}\, b=x \Leftrightarrow b=10^{x}$$

The logarithm base e is called the natural logarithm.

$$\mathrm{ln}\, b=\mathrm{log}_{e}b=x \Leftrightarrow b=e^{x}, \textrm{where}\; e=2,718281828...$$

$$\mathrm{ln}\, b=\int_{1}^{b}\frac{1}{x}\mathrm{d}x$$

## Properties of logarithm

\begin{align} \mathrm{log}_{a}1&=0 \Rightarrow a^{0}=1\\ \\ \mathrm{log}_{a}a&=1 \Rightarrow a^{1}=a\\ \\ \mathrm{log}_{a}0&\; \textrm{is not determined}\\ \\ \mathrm{log}_{a}b&=\frac{1}{\mathrm{log}_{b}a}\\ \\ \mathrm{log}_{a}b&=\frac{\mathrm{log}_{x}b}{\mathrm{log}_{x}a}\\ \end{align}

## Basic rules for logarithms

\begin{align} \mathrm{log}_{a}(xy)&=\, \mathrm{log}_{a}x+\mathrm{log}_{a}y\\ \\ \mathrm{log}_{a}\left ( \frac{x}{y} \right )&=\, \mathrm{log}_{a}x-\mathrm{log}_{a}y,\\ \\ \mathrm{log}_{a}b^{x}&=x\,\mathrm{log}_{a}b\\ \\ \mathrm{log}_{a}\sqrt[x]{b}&=\frac{1}{x}\,\mathrm{log}_{a}b\\ \\ \mathrm{log}_{a}a^{b}&=b\\ \\ a^{\mathrm{log}_{a}b} &= b\\ \end{align}