Definition: Logarithm

Definition: Logarithm (OpenStax Textbook)

A logarithm base \(b\) of a positive number satisfies the following definition.

For \(x>0\), \(b>0\), and \(b \neq 1\), \(y = \log_b(x)\) is equivalent to \(b^y = x\)

• We read \(\log_b(x)\) as, “the log base \(b\) of \(x\)” or “the logarithm with base \(b\) of \(x\)”.
• The logarithm is the exponent that \(b\) must be raised to get \(x\).
• When we evaluate \(\log_b(x)\) we answer the question, “To what exponent must \(b\) be raised to in order to get \(x\)?”

The key idea is that we can rewrite any logarithm in exponential form, as \[\huge{y = \log_b(x) \Leftrightarrow b^y = x}\]