Does the exponent have to be a whole number?

No. The power rule works for any real exponent: negative ($\log_b \frac{1}{M^2} = -2 \log_b M$), fractional ($\log_b \sqrt[3]{M} = \frac{1}{3} \log_b M$) and even irrational ones.

Are $\log(x^2)$ and $2 \log x$ really the same function?

Only for $x > 0$. The function $\log(x^2)$ also exists for negative $x$ (for example $\log((-10)^2) = \log 100 = 2$), but $2 \log x$ does not. For every $x \neq 0$ the exact statement is $\log(x^2) = 2 \log |x|$. The graph above shows both curves.

When should I expand and when should I condense?

Expand when you want simple pieces, for example to approximate $\log 72$ from $\log 2$ and $\log 3$. Condense when you want one logarithm, for example to solve an equation like $2 \log x - \log 4 = 2$ (see Solving Logarithmic Equations).

Teacher Resources

Power Rule for Logarithms

Q: Why does the power rule work?

A logarithm is an exponent. If $\log_b M = x$, then $M = b^x$ and $M^p = b^{px}$. Raising to the power $p$ multiplies the exponent by $p$, so the logarithm (the exponent) is multiplied by $p$ too.

Learn how the power rule moves an exponent in front of a logarithm, how roots become fractional powers, and how to expand and condense logs with all three rules.

Advanced25 minLesson

Printable worksheet

Definition

The power rule for logarithms says that an exponent inside a logarithm can be moved to the front as a factor:

\log_b(M^p) = p \cdot \log_b M

It works for any real exponent

p

, as long as

M > 0

b > 0

and

b \neq 1

Why it is true: let

x = \log_b M

, so

M = b^x

. Then

M^p = (b^x)^p = b^{px}

, and the logarithm of

b^{px}

is simply

px

. So

\log_b(M^p) = p \cdot x = p \cdot \log_b M

Roots are powers too. Since

\sqrt[n]{M} = M^{1/n}

\log_b \sqrt[n]{M} = \frac{1}{n} \log_b M

Example:

\log_2(8^5) = 5 \cdot \log_2 8 = 5 \cdot 3 = 15

, and

\log \sqrt{1000} = \frac{1}{2} \cdot 3 = \frac{3}{2}

Together with the product rule (

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

) and the quotient rule (

\log_b \frac{M}{N} = \log_b M - \log_b N

), the power rule lets you expand one logarithm into many simple ones, or condense several logarithms into one.

Try it now

Which expression is equal to

\log_3(x^5)

for

x > 0

Worked Examples

Evaluate $\log_2(8^5)$ without a calculator.

Move the exponent to the front

$\log_2(8^5) = 5 \cdot \log_2 8$ → $5 \log_2 8$

Evaluate the simpler logarithm

$2^3 = 8$ , so $\log_2 8 = 3$

Multiply

$5 \cdot 3 = 15$

Answer: $\log_2(8^5) = 15$ . Check: $8^5 = (2^3)^5 = 2^{15}$ .

Common Mistakes

Treating $(\log M)^p$ as $p \log M$

Why it's wrong: The power rule only works when the exponent is on the argument, inside the logarithm. In $(\log M)^p$ the whole logarithm is raised to a power. Example: $(\log 100)^3 = 2^3 = 8$ , but $3 \log 100 = 6$ .

Correct: $\log(M^p) = p \log M$ , while $(\log M)^p$ cannot be simplified with the power rule. Look where the exponent sits before you move it.

Applying the exponent to the whole argument: $\log(3x^2) = 2 \log(3x)$

Why it's wrong: In $3x^2$ only $x$ is squared, not $3$ . Check with $x = 10$ : $\log 300 \approx 2.48$ , but $2 \log 30 \approx 2.95$ .

Correct: Split the product first: $\log(3x^2) = \log 3 + \log x^2 = \log 3 + 2 \log x$ .

Turning a square root into a factor of 2: $\log \sqrt{x} = 2 \log x$

Why it's wrong: A square root is the power $\frac{1}{2}$ , not the power $2$ . The index of the root goes into the denominator.

Correct: $\log \sqrt{x} = \log x^{1/2} = \frac{1}{2} \log x$ , and $\log \sqrt[3]{x} = \frac{1}{3} \log x$ .

Condensing $3 \log x - \log y$ into $\log \frac{3x}{y}$

Why it's wrong: A coefficient in front of a logarithm is an exponent of the argument, not a factor of it.

Correct: First move the coefficient up: $3 \log x = \log x^3$ . Then $3 \log x - \log y = \log \frac{x^3}{y}$ .

Interactive Visual

Function graph

Tap or hover over the graph to read off values.

Tap a name to show or hide its graph.

Asymptote

Table of values

x	-10	0.1	1	10	1000
log(x²)	2	-2	0	2	6
2 log(x)	undef.	-2	0	2	6
(log x)²	undef.	1	0	1	9

Interactive Sandbox

Expression Calculator

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Practice Problems

18 problems

Problem 1 of 18

Easy

Which expression is equal to $\log_3(x^5)$ for $x > 0$ ?

Why It Matters

The power rule is the property that makes logarithms truly useful:

Solving exponential equations: in $1.05^t = 2$ the unknown sits in the exponent. Taking the logarithm and using the power rule gives $t \cdot \log 1.05 = \log 2$ , so the exponent comes down and becomes a normal factor.
Huge numbers: $\log(2^{64}) = 64 \log 2 \approx 19.3$ , which tells you at once that $2^{64}$ has 20 digits, without computing it.
Science scales: decibels, pH and the Richter scale all use logarithms; the power rule explains why squaring an intensity ratio doubles the change in decibels.
Calculus: before you differentiate a complicated product of powers, you expand its logarithm with the power rule.

It is the last of the three main log rules, and you will use all three together in almost every logarithm problem from now on.

Real World Applications

Loudness in Decibels

The change in sound level is $\Delta L = 10 \log \frac{I_2}{I_1}$ decibels, where $\frac{I_2}{I_1}$ is how many times the intensity grows. The power rule turns powers of 10 into simple multiplication.

Example:

Doubling your distance from a speaker divides the intensity by $2^2 = 4$ , so $\Delta L = 10 \log(2^{-2}) = -20 \log 2 \approx -6$ dB.

1Try It Yourself

You stand 2 m from a loudspeaker and walk back to 20 m. Sound intensity falls with the square of the distance, so it is divided by $\left(\frac{20}{2}\right)^2$ . The level drops by $10 \log\left(\left(\frac{20}{2}\right)^2\right)$ decibels.

By how many decibels does the sound level drop?

Step 1: Write the mathematical expression

Compute $10 \log(10^2)$ :

Counting Digits of Huge Numbers

A whole number $N$ has $\lfloor \log N \rfloor + 1$ digits. With the power rule you can count the digits of enormous powers without computing them.

Example:

$\log(2^{10}) = 10 \log 2 \approx 3.01$ , so $2^{10} = 1024$ has $3 + 1 = 4$ digits.

2Try It Yourself

Computers often use $64$ -bit numbers, and $2^{64}$ is the number of different values they can store. Use $\log 2 \approx 0.301$ .

How many digits does $2^{64}$ have?

Step 1: Write the mathematical expression

Number of digits: $\lfloor 64 \log 2 \rfloor + 1$

Population Growth

A population that grows by a fixed percentage every year is an exponential function of time. To find when it reaches a target, the power rule brings the time down from the exponent.

Example:

At 10% growth per year, a population doubles when $1.1^t = 2$ : $t \log 1.1 = \log 2$ , so $t = \frac{\log 2}{\log 1.1} \approx 7.3$ years.

3Try It Yourself

A town has 20000 inhabitants and grows by 3% per year. After $t$ years it has $20000 \cdot 1.03^t$ inhabitants.

After how many full years does the town have at least 30000 inhabitants?

Step 1: Write the mathematical expression

Round $t = \frac{\log 1.5}{\log 1.03}$ up to a whole year:

Key Takeaways

1Power rule: $\log_b(M^p) = p \log_b M$ for $M > 0$ and any real exponent $p$
2Roots are fractional powers: $\log_b \sqrt[n]{M} = \frac{1}{n} \log_b M$
3A reciprocal is the power $-1$ : $\log_b \frac{1}{M} = -\log_b M$
4Expanding: split products and quotients first, then bring every exponent to the front
5Condensing: turn every coefficient into an exponent first, then combine sums into products and differences into quotients
6 $(\log M)^p$ is not $p \log M$ : the exponent must be inside the logarithm

Frequently Asked Questions

A logarithm is an exponent. If

\log_b M = x

, then

M = b^x

and

M^p = b^{px}

. Raising to the power

p

multiplies the exponent by

p

, so the logarithm (the exponent) is multiplied by

p

too.

Glossary

Power rule: The logarithm property $\log_b(M^p) = p \log_b M$
Exponent: The number that tells how many times the base is used as a factor, as in $M^p$
Fractional exponent: An exponent written as a fraction; $M^{1/n}$ is the $n$ -th root of $M$
Expand: Rewrite one logarithm as a sum and difference of simpler logarithms
Condense: Rewrite several logarithms as a single logarithm
Coefficient: A number multiplying a logarithm, as the $3$ in $3 \log x$ ; the power rule turns it into an exponent

Formula Card

Power rule

\log_b(M^p) = p \log_b M

The exponent of the argument moves to the front

Root rule

\log_b \sqrt[n]{M} = \frac{1}{n} \log_b M

Because $\sqrt[n]{M} = M^{1/n}$

Reciprocal

\log_b \frac{1}{M} = -\log_b M

Because $\frac{1}{M} = M^{-1}$

Product rule

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

Reminder: products become sums

Quotient rule

\log_b \frac{M}{N} = \log_b M - \log_b N

Reminder: quotients become differences

Power 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

Loudness in Decibels

Counting Digits of Huge Numbers

Population Growth

Key Takeaways

Frequently Asked Questions

Glossary

Formula Card

Power rule

Root rule

Reciprocal

Product rule

Quotient rule

More in This Topic

Introduction to Logarithms

Definition of a Logarithm

Common Logarithms (Base 10)

Natural Logarithms (ln)