Teacher Guide: Power Rule for Logarithms

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.

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10 questions on Logarithms. Students join with a name, you see everyone's score.

For Teachers

Learning Objectives

State the power rule $\log_b(M^p) = p \log_b M$ and explain why it follows from exponent rules
Rewrite logarithms of roots using fractional exponents
Expand logarithmic expressions using the product, quotient and power rules together
Condense sums and differences of logarithms into a single logarithm
Distinguish $\log(M^p)$ from $(\log M)^p$

Prerequisites

• Definition of a Logarithm
• Product Rule for Logarithms
• Quotient Rule for Logarithms
• Exponent rules, including $(b^x)^p = b^{px}$ and $\sqrt[n]{M} = M^{1/n}$

Discussion Starters

1. If a logarithm is an exponent, what happens to it when you raise the argument to the third power?
2. Why is $\log \sqrt{x}$ half of $\log x$ and not double?
3. How can you know how many digits $2^{100}$ has without computing it?
4. The calculator says $\log(100^3) = 6$ but $(\log 100)^3 = 8$ . Which one follows the power rule, and why?

Common Misconceptions

Believing $(\log x)^2 = 2 \log x$

Remediation: Test with $x = 1000$ : $(\log 1000)^2 = 9$ but $2 \log 1000 = 6$ . Then show the three curves in the graph and let students find where they differ.

Moving an exponent that applies to only one factor ( $\log 5x^3 = 3 \log 5x$ )

Remediation: Have students circle exactly what the exponent belongs to before using any rule. Always apply the product rule first, then the power rule.

Thinking the power rule needs a whole-number exponent

Remediation: Derive the rule from $(b^x)^p = b^{px}$ , which holds for every real $p$ , and practice with roots and negative exponents.

Differentiation Ideas

For Struggling Students:

• Start with powers of the base only: $\log_2(2^5)$ , $\log_2(4^3)$ , $\log 10^7$
• Provide a checklist: product rule, quotient rule, power rule, in that order when expanding
• Verify every rewrite with the calculator before moving on

For On-Level Students:

• Expand and condense expressions with three variables and roots
• Approximate logarithms like $\log 72$ or $\log 48$ from $\log 2$ and $\log 3$
• Explain each step with the name of the rule used

For Advanced Students:

• Prove the power rule from the definition and from exponent laws
• Use the power rule to count the digits of $3^{100}$ or $2^{1000}$
• Explore why $\log(x^2) = 2 \log |x|$ for all $x \neq 0$ with the graph

Standards Alignment

HSF-BF.B.5 (CCSS.MATH.CONTENT.HSF.BF.B.5)
Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems
HSF-LE.A.4 (CCSS.MATH.CONTENT.HSF.LE.A.4)
Express the solution to an exponential model as a logarithm and evaluate it using technology
HSA-SSE.A.2 (CCSS.MATH.CONTENT.HSA.SSE.A.2)
Use the structure of an expression to identify ways to rewrite it

Lesson Resources

visualThree Curves Graph
Compare $y = \log(x^2)$ , $y = 2 \log x$ and $y = (\log x)^2$ and their value tables
activityCalculator Check
Students test claims like $\log 8 = 3 \log 2$ and $\log(100^3) = (\log 100)^3$ in the calculator
worksheetExpand and Condense Relay
Pairs alternate: one expands a logarithm, the partner condenses it back

Lesson Content

Everything students see: definition, examples, common mistakes, applications. Tap to open.

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.

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 $3 \log x - \log y = \log \frac{x^3}{y}$ .

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

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.

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).

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

Teacher Guide: Power Rule for Logarithms

Use this lesson with your class

For Teachers

For Struggling Students:

For On-Level Students:

For Advanced Students:

Lesson Content

Definition

Worked Examples

Common Mistakes

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)