Does it work for the natural logarithm too?

Yes, it works for every valid base, including $e$: $\ln(MN) = \ln M + \ln N$. For example $\ln 2 + \ln 5 = \ln 10$.

Is there a rule for $\log(M + N)$?

No. A logarithm of a sum has no simpler form in general. $\log(6 + 4) = \log 10 = 1$, while $\log 6 + \log 4 \approx 1.380$.

How is this different from the quotient and power rules?

They are siblings of the same idea. The product rule turns multiplication into addition; the Quotient Rule for Logarithms turns division into subtraction; the Power Rule for Logarithms turns a power into multiplication by the exponent.

Teacher Resources

Product Rule for Logarithms

Q: Does the product rule work for more than two factors?

Yes. Apply it again and again: $\log_b(MNP) = \log_b M + \log_b N + \log_b P$. For example $\log 42 = \log 2 + \log 3 + \log 7$.

Learn why the logarithm of a product equals the sum of the logarithms, and use it to expand, condense and evaluate logs.

Advanced25 minLesson

Printable worksheet

Definition

The product rule for logarithms says that the logarithm of a product is the sum of the logarithms of its factors:

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

It holds for any base

b > 0

b \neq 1

, and any positive numbers

M

and

N

Why it works: a logarithm is an exponent. Write

M = b^m

and

N = b^n

, so

\log_b M = m

and

\log_b N = n

. When you multiply powers with the same base, you add the exponents:

MN = b^m \cdot b^n = b^{m+n}

So the exponent of

MN

m + n

, which means

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

Check with numbers:

\log_2(4 \cdot 8) = \log_2 32 = 5

and

\log_2 4 + \log_2 8 = 2 + 3 = 5

Used from left to right, the rule expands a log of a product into a sum. Used from right to left, it condenses a sum of logs (same base) into one logarithm.

Try it now

Which expression is equal to

\log_b(MN)

(with

M, N > 0

Worked Examples

Show that $\log_2(4 \cdot 8) = \log_2 4 + \log_2 8$ .

Evaluate the left side

$4 \cdot 8 = 32$ and $2^5 = 32$ → $\log_2 32 = 5$

Evaluate each log on the right side

$2^2 = 4$ and $2^3 = 8$ → $\log_2 4 = 2$ , $\log_2 8 = 3$

Add and compare

$2 + 3 = 5$ , the same as the left side → $5 = 5$

Answer: Both sides equal $5$ : multiplying $4$ by $8$ adds the exponents $2$ and $3$ .

Common Mistakes

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

Why it's wrong: The rule is about a product inside the log, not a sum. Check: $\log(6 + 4) = \log 10 = 1$ , but $\log 6 + \log 4 \approx 0.778 + 0.602 = 1.380$ .

Correct: Only $\log(MN) = \log M + \log N$ . A log of a sum, like $\log(x + 3)$ , cannot be split.

Multiplying the logs: $\log_b(MN) = \log_b M \cdot \log_b N$

Why it's wrong: Multiplying the numbers adds the exponents, it does not multiply them. Check: $\log_2 4 \cdot \log_2 8 = 2 \cdot 3 = 6$ , but $\log_2 32 = 5$ .

Correct: A product inside becomes a sum outside: $\log_2(4 \cdot 8) = 2 + 3 = 5$ .

Combining logs with different bases: $\log_2 x + \log_3 y = \log_6(xy)$

Why it's wrong: The proof uses $b^m \cdot b^n = b^{m+n}$ , which only works when both powers have the same base $b$ .

Correct: Condense only logs with the same base. $\log_2 x + \log_3 y$ stays as it is (or use the Change of Base Formula first).

Using the rule with negative factors: $\log\big((-2)(-5)\big) = \log(-2) + \log(-5)$

Why it's wrong: $\log 10 = 1$ exists, but $\log(-2)$ and $\log(-5)$ do not: the argument of a logarithm must be positive.

Correct: The rule needs $M > 0$ and $N > 0$ . When you condense, keep the domain of the original logs (for example $x > 0$ ).

Interactive Visual

Function graph

a = 4

Tap or hover over the graph to read off values.

log₂(a) = 2

Tap a name to show or hide its graph.

Asymptote

Table of values

x	1	2	4	8	16
log₂(x)	0	1	2	3	4
log₂(a·x)	2	3	4	5	6
log₂(a) + log₂(x)	2	3	4	5	6

Interactive Sandbox

Expression Calculator

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History

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Practice Problems

18 problems

Problem 1 of 18

Easy

Which expression is equal to $\log_b(MN)$ (with $M, N > 0$ )?

Why It Matters

The product rule turns multiplication into addition, which is the whole reason logarithms were invented:

Calculating by hand: before calculators, people multiplied large numbers by adding their logarithms from a table (and with slide rules, which add lengths).
Evaluating logs: if you know $\log 2 \approx 0.301$ and $\log 3 \approx 0.477$ , you get $\log 6 \approx 0.778$ without a calculator.
Sound and science: when a sound gets 100 times more intense, its decibel level goes up by a fixed amount, because $\log(100 \cdot I) = 2 + \log I$ .
Solving equations: in Solving Logarithmic Equations you will condense $\log_2 x + \log_2(x+2)$ into one logarithm before you can solve.
Computer science: combining independent choices multiplies the possibilities but adds the bits, since $\log_2(16 \cdot 64) = 4 + 6$ .

Real World Applications

Decibels and Sound Intensity

Sound level in decibels is $L = 10 \log\left(\frac{I}{I_0}\right)$. When the intensity $I$ is multiplied by a factor $k$, the product rule gives $\log(k \cdot \frac{I}{I_0}) = \log k + \log\frac{I}{I_0}$, so the level goes up by $10 \log k$.

Example:

A sound that becomes 10 times more intense goes up by $10 \log 10 = 10$ decibels, whatever level it started at.

1Try It Yourself

A street measures $60$ decibels. A construction machine makes the sound intensity $1000$ times greater.

What is the new sound level in decibels?

Step 1: Write the mathematical expression

Add $10 \log(1000)$ to the old level:

Slide Rules: Multiplying by Adding

A slide rule marks each number $x$ at a distance $\log x$ from the start. Sliding one scale along the other adds two distances, and by the product rule the sum of the distances is the log of the product.

Example:

To multiply $2 \cdot 3$ , add $\log 2 + \log 3 \approx 0.301 + 0.477 = 0.778$ , which is where $6$ is marked.

2Try It Yourself

On a slide rule, $4$ is at distance $\log 4 \approx 0.602$ and $5$ is at distance $\log 5 \approx 0.699$ . Placing them end to end gives a total distance.

Which product does the slide rule show?

Step 1: Write the mathematical expression

Find the number whose log is the total distance:

Bits of Information

In computer science, the number of bits needed to store one of $N$ equally likely options is $\log_2 N$. Combining two independent choices multiplies the options but adds the bits.

Example:

A choice from $8$ symbols needs $\log_2 8 = 3$ bits, a choice from $4$ needs $2$ bits, and both together have $32$ options: $\log_2 32 = 3 + 2 = 5$ bits.

3Try It Yourself

A game character is saved as one of $16$ colors and one of $64$ shapes.

How many bits are needed to store the color and shape together?

Step 1: Write the mathematical expression

Use $\log_2(16 \cdot 64) = \log_2 16 + \log_2 64$ :

Key Takeaways

1Product rule: $\log_b(MN) = \log_b M + \log_b N$ for $M > 0$ , $N > 0$ , $b > 0$ , $b \neq 1$
2It comes from the exponent rule $b^m \cdot b^n = b^{m+n}$ : multiplying numbers adds their exponents
3Expanding: split a log of a product into a sum, like $\log_3(9x) = 2 + \log_3 x$
4Condensing: combine a sum of logs with the same base into one log, like $\log_4 2 + \log_4 8 = \log_4 16$
5With $\log 2 \approx 0.301$ and $\log 3 \approx 0.477$ you can approximate logs of products such as $\log 6 \approx 0.778$
6There is no rule for $\log(M + N)$ : a log of a sum cannot be split

Frequently Asked Questions

Yes. Apply it again and again:

\log_b(MNP) = \log_b M + \log_b N + \log_b P

. For example

\log 42 = \log 2 + \log 3 + \log 7

Glossary

Product rule: $\log_b(MN) = \log_b M + \log_b N$ : the log of a product is the sum of the logs
Expand: Rewrite one logarithm as a sum of simpler logarithms, like $\log(2x) = \log 2 + \log x$
Condense: Combine a sum of logarithms with the same base into a single logarithm
Argument: The number inside a logarithm, $M$ in $\log_b M$ ; it must be positive
Common logarithm: The logarithm with base 10, written $\log x$

Formula Card

Product rule

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

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

Where it comes from

b^m \cdot b^n = b^{m+n}

Multiplying powers adds the exponents

Three factors

\log_b(MNP) = \log_b M + \log_b N + \log_b P

Apply the rule repeatedly

Useful values

\log 2 \approx 0.301

\log 3 \approx 0.477

Common logarithms for estimating logs of products

Product Rule for Logarithms

Definition

Worked Examples

Common Mistakes

Interactive Visual

Function graph

Table of values

Interactive Sandbox

Expression Calculator

History

Practice Problems

Why It Matters

Real World Applications

Decibels and Sound Intensity

Slide Rules: Multiplying by Adding

Bits of Information

Key Takeaways

Frequently Asked Questions

Glossary

Formula Card

Product rule

Where it comes from

Three factors

Useful values

More in This Topic

Introduction to Logarithms

Definition of a Logarithm

Common Logarithms (Base 10)

Natural Logarithms (ln)