Back to Lesson

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.

Present LessonShare to ClassroomView Student Page

Use this lesson with your class

Free, no student accounts needed.

Share with students

Students open the lesson and practise with instant feedback.

Printable worksheet

All practice problems on paper, with a separate answer key.

Worksheet Answer key

Class quiz

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

For Teachers

Learning Objectives
  • State the power rule log⁡b(Mp)=plog⁡bM\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⁡(Mp)\log(M^p) from (log⁡M)p(\log M)^p
Prerequisites
  • • Definition of a Logarithm
  • • Product Rule for Logarithms
  • • Quotient Rule for Logarithms
  • • Exponent rules, including (bx)p=bpx(b^x)^p = b^{px} and Mn=M1/n\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⁡x\log \sqrt{x} half of log⁡x\log x and not double?
  • 3. How can you know how many digits 21002^{100} has without computing it?
  • 4. The calculator says log⁡(1003)=6\log(100^3) = 6 but (log⁡100)3=8(\log 100)^3 = 8. Which one follows the power rule, and why?
Common Misconceptions

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

Remediation: Test with x=1000x = 1000: (log⁡1000)2=9(\log 1000)^2 = 9 but 2log⁡1000=62 \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⁡5x3=3log⁡5x\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 (bx)p=bpx(b^x)^p = b^{px}, which holds for every real pp, and practice with roots and negative exponents.

Differentiation Ideas

For Struggling Students:

  • • Start with powers of the base only: log⁡2(25)\log_2(2^5), log⁡2(43)\log_2(4^3), log⁡107\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\log 72 or log⁡48\log 48 from log⁡2\log 2 and log⁡3\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 31003^{100} or 210002^{1000}
  • • Explore why log⁡(x2)=2log⁡∣x∣\log(x^2) = 2 \log |x| for all x≠0x \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⁡(x2)y = \log(x^2), y=2log⁡xy = 2 \log x and y=(log⁡x)2y = (\log x)^2 and their value tables

  • activityCalculator Check

    Students test claims like log⁡8=3log⁡2\log 8 = 3 \log 2 and log⁡(1003)=(log⁡100)3\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(Mp)=p⋅log⁡bM\log_b(M^p) = p \cdot \log_b M
It works for any real exponent pp, as long as M>0M > 0, b>0b > 0 and b≠1b \neq 1.
Why it is true: let x=log⁡bMx = \log_b M, so M=bxM = b^x. Then Mp=(bx)p=bpxM^p = (b^x)^p = b^{px}, and the logarithm of bpxb^{px} is simply pxpx. So log⁡b(Mp)=p⋅x=p⋅log⁡bM\log_b(M^p) = p \cdot x = p \cdot \log_b M.
Roots are powers too. Since Mn=M1/n\sqrt[n]{M} = M^{1/n}:
log⁡bMn=1nlog⁡bM\log_b \sqrt[n]{M} = \frac{1}{n} \log_b M
Example: log⁡2(85)=5⋅log⁡28=5⋅3=15\log_2(8^5) = 5 \cdot \log_2 8 = 5 \cdot 3 = 15, and log⁡1000=12⋅3=32\log \sqrt{1000} = \frac{1}{2} \cdot 3 = \frac{3}{2}.
Together with the product rule (log⁡b(MN)=log⁡bM+log⁡bN\log_b(MN) = \log_b M + \log_b N) and the quotient rule (log⁡bMN=log⁡bM−log⁡bN\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(85)\log_2(8^5) without a calculator.
1

Move the exponent to the front

log⁡2(85)=5⋅log⁡28\log_2(8^5) = 5 \cdot \log_2 8 → 5log⁡285 \log_2 8
2

Evaluate the simpler logarithm

23=82^3 = 8, so log⁡28=3\log_2 8 = 3
3

Multiply

5⋅3=155 \cdot 3 = 15

Common Mistakes

Treating (log⁡M)p(\log M)^p as plog⁡Mp \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(\log M)^p the whole logarithm is raised to a power. Example: (log⁡100)3=23=8(\log 100)^3 = 2^3 = 8, but 3log⁡100=63 \log 100 = 6.

Correct: log⁡(Mp)=plog⁡M\log(M^p) = p \log M, while (log⁡M)p(\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⁡(3x2)=2log⁡(3x)\log(3x^2) = 2 \log(3x)

Why it's wrong: In 3x23x^2 only xx is squared, not 33. Check with x=10x = 10: log⁡300≈2.48\log 300 \approx 2.48, but 2log⁡30≈2.952 \log 30 \approx 2.95.

Correct: Split the product first: log⁡(3x2)=log⁡3+log⁡x2=log⁡3+2log⁡x\log(3x^2) = \log 3 + \log x^2 = \log 3 + 2 \log x.

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

Why it's wrong: A square root is the power 12\frac{1}{2}, not the power 22. The index of the root goes into the denominator.

Correct: log⁡x=log⁡x1/2=12log⁡x\log \sqrt{x} = \log x^{1/2} = \frac{1}{2} \log x, and log⁡x3=13log⁡x\log \sqrt[3]{x} = \frac{1}{3} \log x.

Condensing 3log⁡x−log⁡y3 \log x - \log y into log⁡3xy\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: 3log⁡x=log⁡x33 \log x = \log x^3. Then 3log⁡x−log⁡y=log⁡x3y3 \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 1.05t=21.05^t = 2 the unknown sits in the exponent. Taking the logarithm and using the power rule gives t⋅log⁡1.05=log⁡2t \cdot \log 1.05 = \log 2, so the exponent comes down and becomes a normal factor.
  • Huge numbers: log⁡(264)=64log⁡2≈19.3\log(2^{64}) = 64 \log 2 \approx 19.3, which tells you at once that 2642^{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 22=42^2 = 4, so ΔL=10log⁡(2−2)=−20log⁡2≈−6\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 (202)2\left(\frac{20}{2}\right)^2. The level drops by 10log⁡((202)2)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 10log⁡(102)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⁡(210)=10log⁡2≈3.01\log(2^{10}) = 10 \log 2 \approx 3.01, so 210=10242^{10} = 1024 has 3+1=43 + 1 = 4 digits.

2Try It Yourself

Computers often use 6464-bit numbers, and 2642^{64} is the number of different values they can store. Use log⁡2≈0.301\log 2 \approx 0.301.

How many digits does 2642^{64} have?

Step 1: Write the mathematical expression

Number of digits: ⌊64log⁡2⌋+1\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 1.1t=21.1^t = 2: tlog⁡1.1=log⁡2t \log 1.1 = \log 2, so t=log⁡2log⁡1.1≈7.3t = \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 tt years it has 20000⋅1.03t20000 \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=log⁡1.5log⁡1.03t = \frac{\log 1.5}{\log 1.03} up to a whole year:

Key Takeaways

  • 1Power rule: log⁡b(Mp)=plog⁡bM\log_b(M^p) = p \log_b M for M>0M > 0 and any real exponent pp
  • 2Roots are fractional powers: log⁡bMn=1nlog⁡bM\log_b \sqrt[n]{M} = \frac{1}{n} \log_b M
  • 3A reciprocal is the power −1-1: log⁡b1M=−log⁡bM\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(\log M)^p is not plog⁡Mp \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⁡bM=x\log_b M = x, then M=bxM = b^x and Mp=bpxM^p = b^{px}. Raising to the power pp multiplies the exponent by pp, so the logarithm (the exponent) is multiplied by pp too.

Does the exponent have to be a whole number?

No. The power rule works for any real exponent: negative (log⁡b1M2=−2log⁡bM\log_b \frac{1}{M^2} = -2 \log_b M), fractional (log⁡bM3=13log⁡bM\log_b \sqrt[3]{M} = \frac{1}{3} \log_b M) and even irrational ones.

Are log⁡(x2)\log(x^2) and 2log⁡x2 \log x really the same function?

Only for x>0x > 0. The function log⁡(x2)\log(x^2) also exists for negative xx (for example log⁡((−10)2)=log⁡100=2\log((-10)^2) = \log 100 = 2), but 2log⁡x2 \log x does not. For every x≠0x \neq 0 the exact statement is log⁡(x2)=2log⁡∣x∣\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\log 72 from log⁡2\log 2 and log⁡3\log 3. Condense when you want one logarithm, for example to solve an equation like 2log⁡x−log⁡4=22 \log x - \log 4 = 2 (see Solving Logarithmic Equations).

Glossary

Power rule
The logarithm property log⁡b(Mp)=plog⁡bM\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 MpM^p
Fractional exponent
An exponent written as a fraction; M1/nM^{1/n} is the nn-th root of MM
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 33 in 3log⁡x3 \log x; the power rule turns it into an exponent

Formula Card

Power rule

log⁡b(Mp)=plog⁡bM\log_b(M^p) = p \log_b M

The exponent of the argument moves to the front

Root rule

log⁡bMn=1nlog⁡bM\log_b \sqrt[n]{M} = \frac{1}{n} \log_b M

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

Reciprocal

log⁡b1M=−log⁡bM\log_b \frac{1}{M} = -\log_b M

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

Product rule

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

Reminder: products become sums

Quotient rule

log⁡bMN=log⁡bM−log⁡bN\log_b \frac{M}{N} = \log_b M - \log_b N

Reminder: quotients become differences

More in This Topic

Introduction to Logarithms

Advanced25 min

Definition of a Logarithm

Advanced25 min

Common Logarithms (Base 10)

Advanced25 min

Natural Logarithms (ln)

Advanced25 min