Logarithms

Definition

Logarithm

y = \log_{b} x \Leftrightarrow b^{y} = x

b > 0, b ≠ 1, x > 0

Common and natural logs

\log x = \log_{10} x \ln x = \log_{e} x

Log of 1 and of the base

\log_{b} 1 = 0 \log_{b} b = 1

Inverse rules

\log_{b} (b^{x}) = x b^{\log_{b} x} = x

Product rule

\log_{b} (M N) = \log_{b} M + \log_{b} N

Quotient rule

\log_{b} \frac{M}{N} = \log_{b} M - \log_{b} N

Power rule

\log_{b} (M^{k}) = k \log_{b} M

Change of base

\log_{b} a = \frac{\log_{c} a}{\log_{c} b} = \frac{\ln a}{\ln b}

Swap base and argument

\log_{b} a = \frac{1}{\log_{a} b}

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