Power Rule of Logarithms Practice Test

Advanced Algebra Practice Test: ACT math skills.

Power Rule of Logarithms Practice Test

This test has 20 questions

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Power Rule of Logarithms

Raise inside. Multiply outside.

The power rule moves an exponent through a logarithm and turns it into a coefficient. Use the lift in either direction to expand powers, condense coefficients, evaluate expressions, and solve equations.

Read the exponentPreserve the baseCheck the domain

One exponentIdentify the power on the complete argument.

One coefficientMove that exponent in front of the logarithm.

One domainKeep the restrictions from the original form.

The power rule works in both directions

An exponent on the entire positive argument becomes a multiplier of the logarithm value.

Core identity

\log_{b} (M^{r}) = r \log_{b} M

Valid base

b > 0 and b \neq 1

Positive input

M > 0

Real exponent

r is a real number

Scope check: the exponent must apply to the complete logarithm argument, not merely to one unrelated term nearby.

Why the exponent becomes a multiplier

Logarithms report exponents, so raising a power multiplies the reported exponent.

Structural reason

Name the logarithm value

\log_{b} M = p

M = b^{p}

The positive argument is one power of the base.

Raise to a power

Multiply the exponents

M^{r} = {(b^{p})}^{r} = b^{p r}

\log_{b} (M^{r}) = p r = r \log_{b} M

Use familiar powers to evaluate exactly

Move the exponent first, then evaluate the simpler logarithm.

Exact values

Integer exponent

\log_{2} (8^{4})

4 \log_{2} 8 = 4 \cdot 3 = 12

The exponent four multiplies the simpler logarithm value three.

Square-root exponent

\log_{3} \sqrt{27}

\frac{1}{2} \log_{3} 27 = \frac{1}{2} \cdot 3 = \frac{3}{2}

A square root represents an exponent of one half.

Recognize negative and fractional exponents

The same rule handles reciprocals and roots without introducing a new logarithm law.

Power spectrum

Positive integer

\log_{b} (M^{4}) = 4 \log_{b} M

The power becomes a positive coefficient.

Negative integer

\log_{b} (M^{- 2}) = - 2 \log_{b} M

A reciprocal power creates a negative coefficient.

Square root

\log_{b} \sqrt{M} = \frac{1}{2} \log_{b} M

The root becomes a fractional coefficient.

General root

\log_{b} \sqrt[n]{M} = \frac{1}{n} \log_{b} M

The root index becomes the coefficient denominator.

Expand products and quotients one structure at a time

Separate the main factors first, then move every factor exponent outward.

Full expansion

\log (\frac{a^{4} \sqrt{b}}{c^{2}})

Split quotient

\log (a^{4} \sqrt{b}) - \log (c^{2})

The denominator logarithm is subtracted.

Split product

\log (a^{4}) + \log \sqrt{b} - \log (c^{2})

The numerator factors become added logs.

Move powers

4 \log a + \frac{1}{2} \log b - 2 \log c

Each exponent becomes its own coefficient.

Conditions

a > 0, b > 0, c > 0

Positive variables make every expanded logarithm valid.

Move coefficients inward before combining logarithms

The power rule creates powers; product and quotient rules then organize them.

Reverse lift

3 \log x + \log z - 2 \log y

Move inward

\log x^{3} + \log z - \log y^{2}

Every coefficient becomes an exponent.

Build product

\log (x^{3} z) - \log y^{2}

Added terms form a numerator product.

Build quotient

\log (\frac{x^{3} z}{y^{2}})

The subtracted term becomes the denominator.

Domain

x > 0, y > 0, z > 0

Keep all original positivity restrictions.

Use the power rule to expose a simpler equation

Move the coefficient or exponent, convert to exponential form, and verify the domain.

Worked equation

Starting equation

2 \log_{3} x = 4

The original logarithm requires a positive value of the variable.

Divide before converting

\log_{3} x = 2

x = 3^{2} = 9

✓

Verification:

x = 9

is positive, and twice its base-three logarithm equals four.

Even powers require a careful domain check

A power may make an original logarithm valid at negative values even though the expanded logarithm is not.

Domain lens

Do not silently enlarge an identity

\log_{2} (x^{2})

The original expression is defined whenever the variable is not zero.

2 \log_{2} x

This expanded form requires a positive variable. Therefore, the ordinary power-rule rewrite is used with the positive-input condition stated.

Broader real-number form: for a nonzero real variable,

\log_{b} (x^{2}) = 2 \log_{b} | x |

. The absolute value preserves both allowed signs.

Distinguish a powered argument from a powered logarithm value

Parentheses tell you what the exponent actually affects.

Scope test

Exponent inside the logarithm

\log_{b} (M^{3}) = 3 \log_{b} M

The power rule applies because the logarithm receives a powered argument.

Exponent outside the logarithm value

{(\log_{b} M)}^{3} \neq 3 \log_{b} M

Here the logarithm value itself is cubed, so the power rule does not move that exponent.

Reading habit: identify the base, argument, and exponent before choosing a rule.

Know the boundaries of the rule

The power rule moves exponents; it does not distribute logarithms across ordinary sums.

Nonexamples

Added term inside

\log_{b} (M^{2} + 1)

The exponent belongs only to one term, so it cannot move in front of the whole logarithm.

Exponent on the base

\log_{b^{2}} M

This changes the logarithm base and calls for change-of-base reasoning, not the ordinary power rule.

Coefficient inside a sum

\log_{b} (3 M + N)

The number three is not an exponent on the complete argument.

Skills Covered and How to Approach

Separate recognition, transformation, simplification, and domain verification.

Study observatory

Observation sequence

Confirm what the exponent controls and record the original domain before transforming the expression.

\log_{b} (M^{r}) = r \log_{b} M

Then simplify only as far as the question requests and verify the final form.

Skills Covered

Core abilities practiced on this page.

Recognize powered argumentsLocate the exponent attached to the full input.

Expand powers and rootsConvert exponents into coefficients.

Condense coefficientsMove multipliers inward as exponents.

Combine logarithm rulesOrganize products and quotients after moving powers.

Protect the domainKeep restrictions from the original expression.

How to Approach

A five-step observation routine.

Identify the targetDecide whether to expand, condense, evaluate, or solve.

Mark the full argumentCheck exactly what is raised to a power.

Write the domainRequire each original logarithm input to be positive.

Move the exponentPreserve its sign and fractional value.

Simplify and verifyCheck the requested form and original restrictions.

Common Mistakes

Most errors come from reading the exponent incorrectly or forgetting the original domain.

Correction lens

Leaving the exponent inside and outside

Moving the exponent replaces its original position.

Moving only part of an exponent

Preserve the complete signed or fractional exponent.

Applying the rule to a sum

An exponent on one term cannot control the entire argument.

Confusing a square with a coefficient

A powered log value is different from a log of a powered argument.

Forgetting root exponents

Rewrite a root as a fractional power before moving it.

Losing a negative sign

A negative exponent creates a negative coefficient.

Changing the logarithm base

The power rule does not alter the base.

Expanding beyond the domain

State positive-variable assumptions and check solutions.

Final observation audit

Inspect scope, coefficient, base, simplification, and domain before releasing an answer.

Ready check

Exponent lift aligned

A correct power-rule transformation preserves the complete exponent, the logarithm base, and every restriction from the original expression.

Scope · Power · Base · Domain

Does the exponent apply to the complete argument?Check parentheses before moving anything.

Did the full exponent become the coefficient?Keep negative signs and fractions intact.

Did the logarithm base stay unchanged?The power rule moves only the exponent.

Did you apply product or quotient rules afterward?Finish the requested expansion or condensation.

Does the result respect the original domain?Test solutions and preserve stated assumptions.

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