A logarithm is a quantity representing the power to which a fixed number (the base) must be raised to produce a given number.

## Logarithms

A logarithm has base $$a$$ ($$a > 0, a \ne 1$$) of number $$x$$ ($$x \ge 0$$) and is the number $$y$$ such that $$a^y=x$$, thus:

$$y=\log_a{x}$$ means that $$x=a^y$$

## Antilogarithms

An antilogarithm has a logarithm equal to a given number such that:

Where $$\log_a{x}=y$$, $$\textrm{antilog}_a{y}=x$$   ∴ $$x=a^y$$

## Cologarithms

A cologarithm is the logarithm of the reciprocal of a number:

$$\textrm{colog}_a{x} = \log_a{\frac{1}{x}} = -\log_a{x}$$

## Logarithmic Identities

These useful identities will help when working problems involving logarithms.

$$\log{x} = \log_{10}{x}$$ (common logarithm)
$$\ln{x} = \log_{\textrm{ε}}{x}$$ (natural logarithm, ε ≅ 2.71828)
$$\log_a{x^n} = n\,\log_a{x}$$
$$\log_a{x} = \frac{1}{\log_x{a}} = \frac{\log_m{x}}{\log_m{a}}$$
$$\log_a{xy} = \log_a{x}+\log_a{y}$$
$$\log_a{\frac{x}{y}} = \log_a{x}-\log_a{y}$$
$$\log_a{nx^y} = \log_a{n}+y\,\log_a{x}$$
$$\log_a{\sqrt[n]{x}} = \frac{1}{n}\log_a{x}$$
$$\log_a{a^n} = n$$
$$\log_a{\frac{1}{n}} = -\log_a{n}$$
$$a^{\log_a{n}} = n$$
$$a^{x\log_a{n}} = n^x$$
$$a^{\frac{\log_a{x}}{n}} = \sqrt[n]{x}$$
$$\log_a{1} = 0$$
$$\log_a{a} = 1$$
$$\log_a{0} = -\textrm{∞}$$
$$\log_a{\textrm{∞}} = \textrm{∞}$$

## Epsilon (ε, Base of Natural Logarithm)

Also known as the Naperian Constant, the value ε is extremely important in mathematics.

ε ≅ 2.718281828 4590452353 6028747135 2662497757 2470936996