Algebra: Logarithms

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