## Definitions

### General

The logarithm of a number x to a base n, is a power p so that n^{p} = x. In modern notation logarithm is written as \(\log_{n}{x}\). Any positive real number has a logarithm to a positive real base (except base 1). For any valid base, logarithm of 1 is 0, while the logarithm of 0 approaches negative infinity. Changing between bases can be done with the formula:

\[ \log_{n} x = \frac{\log_{m}{x}}{\log_{m}{n}} \]

### Common bases

In practice, some logarithmic bases find more frequent use than others. Such common bases are 10 (decimal logarithm, \(\log_{10}{x}\) or \(\lg{x}\)), 2 (binary logarithm, \(\log_{2}{x}\)) and the mathematical constant e≈2.718, (natural logarithm, \(\ln{x}\) )

### Properties

The derivative of the logarithm function is:

\[ \left(\log_{n} x\right)' = \frac{1}{x}\frac{1}{\ln{n}} \]

The integral of the decimal logarithm function is given by:

\[ \int \log_{n} x\, \mathrm{d}x = \frac{x\ln{x}-x}{\ln{n}} +C \]

Identities:

\[ \begin{split} & \log_{n} \left(x y\right) & = \log_{n}{x} + \log_{n}{y} \\ \\ & \log_{n} \left(\frac{x}{y}\right) & = \log_{n}{x} - \log_{n}{y} \\ \\ & \log_{n} \left(x^p\right) & = p\log_{n}{x} \\ \\ & \log_{n} \left(\sqrt[p]{x}\right) & = \frac{\log_{n}{x}}{p} \\ \\ & \log_{n} {n} & = 1 \\ \\ & \log_{n} {n^x} & = x \\ \\ \end{split} \]