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}
See also:
FAQ
What is a logarithm?
A logarithm is the exponent you raise a base to in order to get a given number.
What is the difference between common and natural logarithms?
Common logarithms use base 10, while natural logarithms use base e.
Why are logarithms useful?
They simplify multiplication, division, and power relationships in algebra and calculus.
