Logarithm power rule

What the symbols mean

$L$

the log of the power, ${\log }_b(M^p)$

$b$

the base (positive, not 1)

$M$

the number being raised to a power (positive)

$p$

the power (any real number)

When to use it

Use it whenever the unknown is in an exponent: take logs of both sides, then bring the exponent down. For example $3^x=20$ becomes $x\log 3=\log 20$. It also turns roots into fractions: $\log \sqrt{M}=\frac{12}{\log }M$.

Why it works

$M^p$ is p copies of M multiplied together, and by the product rule each copy adds ${\log }_{b}M$. For non-whole p, write $M=b^m$: then $M^p=b^{mp}$.

Common mistakes

• Applying it to $(\log M{)}^p$. Only an exponent *inside* the log comes out: $(\log M{)}^2\ne 2\log M$.
• Bringing out an exponent that applies to only part of the argument: $\log (3x^2)=\log 3+2\log x$, not $2\log (3x)$.