The Quotient Rule

Idea

Subtraction is the product rule read the other way. Two logarithms with the same base are subtracted by dividing their arguments.

Exponents subtract when powers divide, and a logarithm is an exponent. That is the whole content of the rule.

Rule

$\log_a x - \log_a y = \log_a \dfrac{x}{y}$

$\log_a \dfrac{1}{y} = -\log_a y$

How it is used

1. Where it comes from

\log_a x = m \;\Rightarrow\; a^m = x,

\log_a y = n \;\Rightarrow\; a^n = y

\frac{x}{y} = \frac{a^m}{a^n} = a^{m-n} \;\Rightarrow\; \log_a \frac{x}{y} = m - n

And $m - n$ is $\log_a x - \log_a y$ .

2. Subtracting to get a value you can read

\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

The order matters. $\log_a x - \log_a y$ puts $x$ on top.

3. A reciprocal flips the sign

Put $x = 1$ in the rule. Since $\log_a 1 = 0$ :

\log_a \frac{1}{y} = 0 - \log_a y = -\log_a y

So $\log_2 \dfrac{1}{8} = -3$ and $\ln \dfrac{1}{e} = -1$ without any extra work.

Exponential and logarithm

Try this. Compare bases 2 and 0.5: growth becomes decay. Move the input to 1; the logarithm is zero for either base.

BasePositive input x0.5

Base 2 · Blue: y = 2ˣ · Green: y = log_b(x) · Dashed: y = x. At x = 0.5, y = -1.

4. Using 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

Expression	Single logarithm
$\log_2 20 - \log_2 5$	$\log_2 4 = 2$
$\log_3 2 + \log_3 6 - \log_3 4$	$\log_3 3 = 1$
$-\log_5 25$	$\log_5 \dfrac{1}{25} = -2$

And what the rule does not say:

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

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

Worked example

Evaluate $\log_2 96 - \log_2 3 + \log_5 \dfrac{1}{125}$ .

Remember

$\log_a x - \log_a y = \log_a \dfrac{x}{y}$ , same base only $\log_a \dfrac{1}{y} = -\log_a y$ work base by base, then add the finished values

The first two share base $2$ . Divide their arguments.

\log_2 96 - \log_2 3 = \log_2 \frac{96}{3} = \log_2 32

Write $32$ as a power of $2$ .

\log_2 32 = \log_2 2^5 = 5

The last term is a reciprocal, so the sign flips.

\log_5 \frac{1}{125} = -\log_5 125 = -3

Add the two finished values.

5 + (-3) = 2

Explore

Drag the probe along the green curve and read the height at two values of $x$ .

With the base at $3$ , the height at $x = 9$ is $2$ and at $x = 3$ it is $1$ . Their difference is $1$ , which is the height at $x = 9 \div 3 = 3$ . Dividing along the bottom subtracts along the side.

Now drag the probe to the left of $x = 1$ and watch the height go negative. Everything between $0$ and $1$ is a reciprocal of something larger, and the rule $\log_a \frac{1}{y} = -\log_a y$ is why that whole stretch of curve sits below the axis.