Logarithm quotient rule

What the symbols mean

$L$

the log of the quotient, ${\log }_b(M/N)$

$b$

the base (positive, not 1)

$M$

the numerator (positive)

$N$

the denominator (positive)

When to use it

Use it to expand the log of a fraction, or to combine a difference of logs into one log when solving equations. It is why decibels and pH differences are ratios: a difference of logs is the log of a ratio.

Why it works

With $M=b^m$ and $N=b^n$, $M/N=b^{m-n}$, so ${\log }_b(M/N)=m-n$.

Common mistakes

• Writing $\log M/\log N$ for $\log (M/N)$. A quotient inside becomes a difference outside, not a division.
• Subtracting in the wrong order: the numerator's log comes first.
• Forgetting that ${\log }_b(1/N)=-{\log }_{b}N$, a special case of this rule with M = 1.