Power Law

For \(a>0,\ a\ne1,\ b>0\) and real \(r\),

\(\log_a(b^r)=r\log_a b\)

Example

\(\log_2(5^x)=x\log_2 5\)

\(\log_3\sqrt2=\dfrac12\log_3 2\)

Take note

The power law applies to a power inside the logarithm. It does not make \((\log_a x)^3=3\log_a x\) an identity. For \(x>0\),

\((\log_a x)^3=(\log_a x)(\log_a x)(\log_a x)\)