Product Rule for Logarithms

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

Advanced25 min18 problems
Practice Problems

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⁡bM+log⁡bN\log_b(MN) = \log_b M + \log_b N
It holds for any base b>0b > 0, b≠1b \neq 1, and any positive numbers MM and NN.
Why it works: a logarithm is an exponent. Write M=bmM = b^m and N=bnN = b^n, so log⁡bM=m\log_b M = m and log⁡bN=n\log_b N = n. When you multiply powers with the same base, you add the exponents:
MN=bm⋅bn=bm+nMN = b^m \cdot b^n = b^{m+n}
So the exponent of MNMN is m+nm + n, which means log⁡b(MN)=m+n=log⁡bM+log⁡bN\log_b(MN) = m + n = \log_b M + \log_b N.
Check with numbers: log⁡2(4⋅8)=log⁡232=5\log_2(4 \cdot 8) = \log_2 32 = 5 and log⁡24+log⁡28=2+3=5\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)\log_b(MN) (with M,N>0M, N > 0)?

Worked Examples

Show that log⁡2(4⋅8)=log⁡24+log⁡28\log_2(4 \cdot 8) = \log_2 4 + \log_2 8.
1

Evaluate the left side

4⋅8=324 \cdot 8 = 32 and 25=322^5 = 32 → log⁡232=5\log_2 32 = 5
2

Evaluate each log on the right side

22=42^2 = 4 and 23=82^3 = 8 → log⁡24=2\log_2 4 = 2, log⁡28=3\log_2 8 = 3
3

Add and compare

2+3=52 + 3 = 5, the same as the left side → 5=55 = 5

Common Mistakes

Writing log⁡(M+N)=log⁡M+log⁡N\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\log(6 + 4) = \log 10 = 1, but log⁡6+log⁡4≈0.778+0.602=1.380\log 6 + \log 4 \approx 0.778 + 0.602 = 1.380.

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

Multiplying the logs: log⁡b(MN)=log⁡bM⋅log⁡bN\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⁡24⋅log⁡28=2⋅3=6\log_2 4 \cdot \log_2 8 = 2 \cdot 3 = 6, but log⁡232=5\log_2 32 = 5.

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

Combining logs with different bases: log⁡2x+log⁡3y=log⁡6(xy)\log_2 x + \log_3 y = \log_6(xy)

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

Correct: Condense only logs with the same base. log⁡2x+log⁡3y\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⁡((−2)(−5))=log⁡(−2)+log⁡(−5)\log\big((-2)(-5)\big) = \log(-2) + \log(-5)

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

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

Interactive Visual

Function graph

051015-202468xyx = 0

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  • log₂(a) = 2

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Asymptote

Table of values

x124816
log₂(x)01234
log₂(a·x)23456
log₂(a) + log₂(x)23456

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Practice Problems

18 problems
Problem 1 of 18
Easy
Which expression is equal to log⁡b(MN)\log_b(MN) (with M,N>0M, 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≈0.301\log 2 \approx 0.301 and log⁡3≈0.477\log 3 \approx 0.477, you get log⁡6≈0.778\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⋅I)=2+log⁡I\log(100 \cdot I) = 2 + \log I.
  • Solving equations: in Solving Logarithmic Equations you will condense log⁡2x+log⁡2(x+2)\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⋅64)=4+6\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 10log⁡10=1010 \log 10 = 10 decibels, whatever level it started at.

1Try It Yourself

A street measures 6060 decibels. A construction machine makes the sound intensity 10001000 times greater.

What is the new sound level in decibels?

Step 1: Write the mathematical expression

Add 10log⁡(1000)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⋅32 \cdot 3, add log⁡2+log⁡3≈0.301+0.477=0.778\log 2 + \log 3 \approx 0.301 + 0.477 = 0.778, which is where 66 is marked.

2Try It Yourself

On a slide rule, 44 is at distance log⁡4≈0.602\log 4 \approx 0.602 and 55 is at distance log⁡5≈0.699\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 88 symbols needs log⁡28=3\log_2 8 = 3 bits, a choice from 44 needs 22 bits, and both together have 3232 options: log⁡232=3+2=5\log_2 32 = 3 + 2 = 5 bits.

3Try It Yourself

A game character is saved as one of 1616 colors and one of 6464 shapes.

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

Step 1: Write the mathematical expression

Use log⁡2(16⋅64)=log⁡216+log⁡264\log_2(16 \cdot 64) = \log_2 16 + \log_2 64:

Key Takeaways

  • 1Product rule: log⁡b(MN)=log⁡bM+log⁡bN\log_b(MN) = \log_b M + \log_b N for M>0M > 0, N>0N > 0, b>0b > 0, b≠1b \neq 1
  • 2It comes from the exponent rule bm⋅bn=bm+nb^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⁡3x\log_3(9x) = 2 + \log_3 x
  • 4Condensing: combine a sum of logs with the same base into one log, like log⁡42+log⁡48=log⁡416\log_4 2 + \log_4 8 = \log_4 16
  • 5With log⁡2≈0.301\log 2 \approx 0.301 and log⁡3≈0.477\log 3 \approx 0.477 you can approximate logs of products such as log⁡6≈0.778\log 6 \approx 0.778
  • 6There is no rule for log⁡(M+N)\log(M + N): a log of a sum cannot be split

Frequently Asked Questions

Yes. Apply it again and again: log⁡b(MNP)=log⁡bM+log⁡bN+log⁡bP\log_b(MNP) = \log_b M + \log_b N + \log_b P. For example log⁡42=log⁡2+log⁡3+log⁡7\log 42 = \log 2 + \log 3 + \log 7.
Yes. Apply it again and again: log⁡b(MNP)=log⁡bM+log⁡bN+log⁡bP\log_b(MNP) = \log_b M + \log_b N + \log_b P. For example log⁡42=log⁡2+log⁡3+log⁡7\log 42 = \log 2 + \log 3 + \log 7.
Yes, it works for every valid base, including ee: ln⁡(MN)=ln⁡M+ln⁡N\ln(MN) = \ln M + \ln N. For example ln⁡2+ln⁡5=ln⁡10\ln 2 + \ln 5 = \ln 10.
No. A logarithm of a sum has no simpler form in general. log⁡(6+4)=log⁡10=1\log(6 + 4) = \log 10 = 1, while log⁡6+log⁡4≈1.380\log 6 + \log 4 \approx 1.380.
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.

Glossary

Product rule
log⁡b(MN)=log⁡bM+log⁡bN\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\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, MM in log⁡bM\log_b M; it must be positive
Common logarithm
The logarithm with base 10, written log⁡x\log x

Formula Card

Product rule

log⁡b(MN)=log⁡bM+log⁡bN\log_b(MN) = \log_b M + \log_b N

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

Where it comes from

bm⋅bn=bm+nb^m \cdot b^n = b^{m+n}

Multiplying powers adds the exponents

Three factors

log⁡b(MNP)=log⁡bM+log⁡bN+log⁡bP\log_b(MNP) = \log_b M + \log_b N + \log_b P

Apply the rule repeatedly

Useful values

log⁡2≈0.301\log 2 \approx 0.301, log⁡3≈0.477\log 3 \approx 0.477

Common logarithms for estimating logs of products

Today's challengeOne problem, three tries

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Common Logarithms (Base 10)

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