Logarithm product rule

What the symbols mean

$L$

the log of the product, ${\log }_b(MN)$

$b$

the base (positive, not 1)

$M$

the first factor (positive)

$N$

the second factor (positive)

When to use it

Use it to split a log of a product into simpler logs, to combine a sum of logs into a single log before solving an equation such as ${\log }_{2}x+{\log }_2(x-2)=3$, and to see why slide rules and log tables worked: they replaced multiplication with addition.

Why it works

If $M=b^m$ and $N=b^n$, then $MN=b^{m+n}$ by the laws of exponents, so ${\log }_b(MN)=m+n={\log }_{b}M+{\log }_{b}N$.

Common mistakes

• Writing $\log (M+N)=\log M+\log N$. The rule is for a product, not a sum; $\log (M+N)$ does not simplify.
• Writing $\log M·\log N$ for $\log (MN)$. Logs of a product add; they do not multiply.
• Combining logs with different bases. The rule needs the same base on every term.