Logarithm with arbitrary base

Summary

• For any given base \(b>0 \) and any given real number \(c>0\) the equation

\[b^a=c\]

• has a unique solution \(a\) .
• The solution \(a\) is denoted by \({log}_b c\) and pronounced the logarithm of \(c\) to base \(b\) .
• For example, \({log}_2 8={log}_2 2^3=3\) and \({log}_4 0.25={log}_4 4^{-1}=-1\) .
• If \(c≤0\) then \({log}_b c\) is not defined.

Logarithmic identities and laws

• \({log}_b b^a=a\) for all \(a\)
• \(b^{{log}_b c}=c\) for \(c>0\)

Logarithm of products, ratios and powers

• \({log}_b cd={log}_b c+{log}_b d\)
• \({log}_b \frac{c}{d}={log}_b c-{log}_b d\)
• \({log}_b c^d=d{log}_b c\)
• \({log}_b c= \frac{ln c}{ln b}= \frac{log c}{log b}\)
• Given \(b^a=c\) where \(c>0\) is given,
• You get the base \(b\) from exponent \(a\) using a radical, \(b=\sqrt[a]{c}\)
• You get the exponent \(a\) from the base \(b\) using a logarithm, \(a={log}_b c\)

Lecture coming soon