The Quotient Rule — Cheat sheet

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The rule

\log_a x - \log_a y = \log_a \frac{x}{y}

Same base only. The first argument goes on top.

Why

Exponents subtract when powers divide.

\frac{a^m}{a^n} = a^{m-n}

The reciprocal shortcut

Put $x = 1$ and use $\log_a 1 = 0$ :

\log_a \frac{1}{y} = -\log_a y

So $\log_2 \dfrac{1}{8} = -3$ , $\log_3 \dfrac{1}{9} = -2$ , $\ln \dfrac{1}{e} = -1$ . This is why the whole curve between $0$ and $1$ sits below the axis.

Worth knowing by sight

Expression	Single logarithm	Value
$\log_2 48 - \log_2 3$	$\log_2 16$	$4$
$\log 5000 - \log 5$	$\log 1000$	$3$
$\ln e^7 - \ln e^4$	$\ln e^3$	$3$
$\log_2 5 - \log_2 40$	$\log_2 \dfrac{1}{8}$	$-3$
$\log_2 20 - \log_2 5$	$\log_2 4$	$2$

Both rules together

Sums go on top, differences go underneath.

\log_a 6 + \log_a 5 - \log_a 3 = \log_a \frac{6 \cdot 5}{3} = \log_a 10

\log_6 4 + \log_6 27 - \log_6 18 = \log_6 \frac{108}{18} = \log_6 6 = 1

Traps

\log_a x - \log_a y \ne \log_a(x - y)

\frac{\log_a x}{\log_a y} \ne \log_a \frac{x}{y}

Order matters: reversing the two terms flips the sign of the answer.

Given-value questions

\log_a 3 = 0.5,\; \log_a 2 = 0.3 \;\Rightarrow\; \log_a 1.5 = \log_a \frac{3}{2} = 0.5 - 0.3 = 0.2