What happens when the quotient is less than 1?

Then the logarithm is negative (for a base greater than 1), and the rule shows why: $\log_2\left(\dfrac{1}{8}\right) = \log_2 1 - \log_2 8 = 0 - 3 = -3$. In general $\log_b\left(\dfrac{1}{N}\right) = -\log_b N$.

Can I expand $\log\left(\dfrac{x}{x - 2}\right)$?

Yes, for $x > 2$: $\log x - \log(x - 2)$. But $\log(x - 2)$ stays as it is, because there is no rule for the logarithm of a difference.

Why is it called the twin of the product rule?

Dividing by $N$ is multiplying by $\dfrac{1}{N}$, so $\log\left(\dfrac{M}{N}\right) = \log M + \log\left(\dfrac{1}{N}\right) = \log M - \log N$. Every quotient rule step is a product rule step in disguise.

Teacher Resources

Quotient Rule for Logarithms

Q: Do both logarithms need the same base?

Yes. $\log_b M - \log_b N = \log_b\left(\dfrac{M}{N}\right)$ only works when both terms have the same base $b$. A difference like $\log_2 8 - \log_3 9$ cannot be condensed into one logarithm with this rule.

Learn why the logarithm of a quotient equals a difference of logarithms, and use the rule to expand and condense expressions.

Advanced25 minLesson

Printable worksheet

Definition

The quotient rule for logarithms says: the logarithm of a quotient is the logarithm of the numerator minus the logarithm of the denominator.

\log_b\left(\dfrac{M}{N}\right) = \log_b M - \log_b N

The rule holds for

M > 0

N > 0

and a base

b > 0

with

b \neq 1

Why it works: a logarithm is an exponent. Write

\log_b M = x

and

\log_b N = y

, so

M = b^x

and

N = b^y

. Dividing powers with the same base subtracts the exponents:

\dfrac{M}{N} = \dfrac{b^x}{b^y} = b^{x-y} \quad\Rightarrow\quad \log_b\left(\dfrac{M}{N}\right) = x - y = \log_b M - \log_b N

It is the twin of the product rule (

\log_b(MN) = \log_b M + \log_b N

): multiplying inside the logarithm becomes adding outside, and dividing inside becomes subtracting outside.

Example:

\log_2\left(\dfrac{32}{4}\right) = \log_2 32 - \log_2 4 = 5 - 2 = 3

, and indeed

\log_2 8 = 3

You can use the rule in two directions:

Expand: split one logarithm of a quotient into a difference, $\log_3\left(\dfrac{x}{7}\right) = \log_3 x - \log_3 7$
Condense: combine a difference into one logarithm, $\log 500 - \log 5 = \log 100 = 2$

Try it now

Which expression is equal to

\log_b\left(\dfrac{M}{N}\right)

(for

M, N > 0

Worked Examples

Show that $\log_2\left(\dfrac{32}{4}\right) = \log_2 32 - \log_2 4$ .

Evaluate the left side directly

$\dfrac{32}{4} = 8$ and $2^3 = 8$ → $\log_2 8 = 3$

Evaluate each logarithm on the right side

$2^5 = 32$ and $2^2 = 4$ → $\log_2 32 = 5$ , $\log_2 4 = 2$

Subtract

$5 - 2 = 3$

Compare both sides

Left side $3$ , right side $3$ → Both sides are equal

Answer: Both sides equal $3$ , so $\log_2\left(\dfrac{32}{4}\right) = \log_2 32 - \log_2 4$ .

Common Mistakes

Writing $\dfrac{\log M}{\log N} = \log\left(\dfrac{M}{N}\right)$

Why it's wrong: A quotient of two logarithms is not the logarithm of a quotient. Test it: $\dfrac{\log 100}{\log 10} = \dfrac{2}{1} = 2$ , but $\log\left(\dfrac{100}{10}\right) = \log 10 = 1$ .

Correct: The rule turns division INSIDE the logarithm into subtraction: $\log\left(\dfrac{M}{N}\right) = \log M - \log N$ . The fraction $\dfrac{\log M}{\log N}$ has no shortcut of this kind.

Writing $\log(M - N) = \log M - \log N$

Why it's wrong: The rule is about a quotient inside the logarithm, not a difference. Test it: $\log(100 - 10) = \log 90 \approx 1.954$ , but $\log 100 - \log 10 = 1$ .

Correct: There is no rule that splits $\log(M - N)$ . Only $\log\left(\dfrac{M}{N}\right)$ becomes $\log M - \log N$ .

Subtracting in the wrong order: $\log_b\left(\dfrac{M}{N}\right) = \log_b N - \log_b M$

Why it's wrong: Subtraction is not commutative, so the order matters. $\log_2\left(\dfrac{8}{2}\right) = \log_2 4 = 2$ , but $\log_2 2 - \log_2 8 = 1 - 3 = -2$ .

Correct: Numerator first, minus denominator: $\log_b\left(\dfrac{M}{N}\right) = \log_b M - \log_b N$ .

Writing $\log\left(\dfrac{M}{NP}\right) = \log M - \log N + \log P$

Why it's wrong: Both $N$ and $P$ are in the denominator, so both logarithms are subtracted. The $+$ treats $P$ as if it were in the numerator.

Correct: $\log\left(\dfrac{M}{NP}\right) = \log M - \log(NP) = \log M - \log N - \log P$ . Use parentheses: $\log M - (\log N + \log P)$ .

Interactive Visual

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a = 2

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Asymptote

Table of values

x	1	2	4	5	10	20
log(x/a)	-0.30103	0	0.30103	0.39794	0.69897	1
log(x) - log(a)	-0.30103	0	0.30103	0.39794	0.69897	1
log(x) / log(a) (WRONG: not equal to log(x/a))	0	1	2	2.32193	3.32193	4.32193

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Practice Problems

18 problems

Problem 1 of 18

Easy

Which expression is equal to $\log_b\left(\dfrac{M}{N}\right)$ (for $M, N > 0$ )?

Why It Matters

The quotient rule turns division into subtraction, and that is exactly what many formulas need:

Comparing two measurements: Decibels, pH and the Richter scale all compare a quantity to a reference value with $\log\left(\dfrac{I}{I_0}\right)$ . The quotient rule shows that the difference between two levels depends only on the ratio of the two quantities.
Sound: Two sounds whose intensities have the ratio $1000$ differ by $10 \log 1000 = 30$ decibels, no matter how loud they are.
Chemistry: Two solutions whose hydrogen ion concentrations differ by a factor of $100$ differ by exactly $2$ on the pH scale.
Simplifying expressions: Expanding $\log\left(\dfrac{ab}{c}\right)$ into $\log a + \log b - \log c$ makes later algebra (and calculus) much easier.
Solving equations: Condensing $\log_3 x - \log_3 4$ into $\log_3\left(\dfrac{x}{4}\right)$ is often the first step in solving a logarithmic equation.

Together with the product rule and the power rule, the quotient rule lets you rewrite almost any logarithmic expression into the form you need.

Real World Applications

Comparing Sound Levels in Decibels

Sound level is $L = 10 \log\left(\dfrac{I}{I_0}\right)$ decibels, where $I$ is the intensity and $I_0$ a fixed reference. By the quotient rule, the difference of two levels is $10(\log I_A - \log I_B) = 10 \log\left(\dfrac{I_A}{I_B}\right)$: the reference $I_0$ cancels.

Example:

A sound 100 times more intense than another is $10 \log 100 = 20$ decibels louder.

1Try It Yourself

A loudspeaker produces an intensity of $I_A = 0.01$ W/m², and a quiet street produces $I_B = 0.00001$ W/m².

How many decibels louder is the loudspeaker than the street?

Step 1: Write the mathematical expression

Calculate $10 \log\left(\dfrac{I_A}{I_B}\right)$ :

Comparing Acidity with pH

Chemists define $\text{pH} = -\log[H^+]$, where $[H^+]$ is the hydrogen ion concentration. By the quotient rule, the pH difference of two solutions is the logarithm of the ratio of their concentrations.

Example:

Coffee has $[H^+] = 10^{-5}$ and pure water $10^{-7}$ . The pH difference is $\log\left(\dfrac{10^{-5}}{10^{-7}}\right) = \log 100 = 2$ (pH 5 versus pH 7).

2Try It Yourself

Tomato juice has $[H^+] = 10^{-4}$ and pure water has $[H^+] = 10^{-7}$ .

By how much is the pH of tomato juice lower than the pH of water?

Step 1: Write the mathematical expression

Calculate $\log\left(\dfrac{10^{-4}}{10^{-7}}\right)$ :

Comparing Earthquakes on the Richter Scale

The Richter magnitude is $M = \log\left(\dfrac{A}{A_0}\right)$, where $A$ is the amplitude of the ground motion and $A_0$ a reference amplitude. For two earthquakes measured at the same distance, the quotient rule gives $M_1 - M_2 = \log\left(\dfrac{A_1}{A_2}\right)$.

Example:

If one earthquake has 10 times the amplitude of another, its magnitude is $\log 10 = 1$ higher.

3Try It Yourself

Two earthquakes are recorded at the same distance. The first has amplitude $A_1 = 50000$ micrometers, the second $A_2 = 500$ micrometers. The second earthquake has magnitude $4$ .

What is the magnitude of the first earthquake?

Step 1: Write the mathematical expression

Calculate $4 + \log\left(\dfrac{A_1}{A_2}\right)$ :

Key Takeaways

1Quotient rule: $\log_b\left(\dfrac{M}{N}\right) = \log_b M - \log_b N$ for $M, N > 0$
2It comes from the exponent rule $\dfrac{b^x}{b^y} = b^{x-y}$ : dividing powers subtracts exponents
3Expanding splits a logarithm of a quotient into a difference; condensing combines a difference into one logarithm
4Numerator first: the logarithm of the denominator is always the one subtracted
5With the product rule: $\log\left(\dfrac{ab}{c}\right) = \log a + \log b - \log c$ and $\log\left(\dfrac{a}{bc}\right) = \log a - \log b - \log c$
6 $\dfrac{\log M}{\log N}$ and $\log(M - N)$ are NOT equal to $\log\left(\dfrac{M}{N}\right)$

Frequently Asked Questions

Yes.

\log_b M - \log_b N = \log_b\left(\dfrac{M}{N}\right)

only works when both terms have the same base

b

. A difference like

\log_2 8 - \log_3 9

cannot be condensed into one logarithm with this rule.

Glossary

Quotient rule: The property $\log_b\left(\dfrac{M}{N}\right) = \log_b M - \log_b N$ for $M, N > 0$
Quotient: The result of a division; in $\dfrac{M}{N}$ , $M$ is the numerator and $N$ the denominator
Expand: Rewrite one logarithm of a product or quotient as a sum or difference of simpler logarithms
Condense: Combine a sum or difference of logarithms with the same base into a single logarithm
Argument: The number inside a logarithm, such as $M$ in $\log_b M$ ; it must be positive

Formula Card

Quotient rule

\log_b\left(\dfrac{M}{N}\right) = \log_b M - \log_b N

For $M > 0$, $N > 0$, $b > 0$, $b \neq 1$

Reciprocal

\log_b\left(\dfrac{1}{N}\right) = -\log_b N

Special case with $M = 1$, because $\log_b 1 = 0$

With the product rule

\log_b\left(\dfrac{MN}{P}\right) = \log_b M + \log_b N - \log_b P

Numerator factors are added, denominator factors subtracted

Not a rule

\dfrac{\log_b M}{\log_b N} \neq \log_b\left(\dfrac{M}{N}\right)

A quotient of logarithms is not the logarithm of a quotient

Quotient Rule for Logarithms

Definition

Worked Examples

Common Mistakes

Interactive Visual

Function graph

Table of values

Interactive Sandbox

Interactive Grapher

Expression Calculator

History

Practice Problems

Why It Matters

Real World Applications

Comparing Sound Levels in Decibels

Comparing Acidity with pH

Comparing Earthquakes on the Richter Scale

Key Takeaways

Frequently Asked Questions

Glossary

Formula Card

Quotient rule

Reciprocal

With the product rule

Not a rule

More in This Topic

Introduction to Logarithms

Definition of a Logarithm

Common Logarithms (Base 10)

Natural Logarithms (ln)