Logarithmic Equations with Exponents Practice Test

Advanced Algebra Practice Test: ACT math skills.

Logarithmic Equations with Exponents Practice Test

This test has 20 questions

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Logarithmic Equations with Exponents

Switch forms. Follow the exponent.

Logarithmic and exponential equations describe the same relationship in two forms. Choose the form that exposes the unknown, solve the resulting algebra, and return to the original domain for a final check.

Write the domain Convert forms Solve and verify

Restrict Every logarithm input must be positive.

Translate Move between logarithmic and exponential form.

Solve Use algebra or logarithms to isolate the variable.

Verify Test candidates in the original equation.

Relay 01

The two forms carry the same information

The base, exponent, and result keep their roles when the equation changes form.

Core equivalence

Read the positions carefully

\log_{b} M = r \Leftrightarrow b^{r} = M

b > 0, b \neq 1, M > 0

The logarithm asks for the exponent that makes the exponential statement true.

Relay 02

Convert a basic logarithmic equation

When the unknown is inside the logarithm argument, exponential form often isolates it immediately.

Direct conversion

Example

\log_{2} (x - 1) = 4

x - 1 = 2^{4} = 16

x = 17

The result satisfies the original requirement that the argument be positive.

Relay 03

Cancel matching logarithm and exponential bases

A logarithm undoes an exponential function when both use the same valid base.

Inverse operations

Expose the exponent

\log_{3} (3^{2 x - 1}) = 5

2 x - 1 = 5

x = 3

The exponential input is always positive, so the logarithm is defined.

Relay 04

Use the power rule before solving

An outside coefficient can be divided away or moved inward as an exponent.

Coefficient route

Choose the shorter route

2 \log_{3} x = 4

Divide first

\log_{3} x = 2

Then convert

x = 3^{2} = 9

A powered argument may create more than one solution

After converting forms, solve the resulting power equation completely.

Two candidates

\log_{2} ({(x - 1)}^{2}) = 4

Domain

x \neq 1

The squared argument must be positive, so it cannot equal zero.

Convert

{(x - 1)}^{2} = 2^{4} = 16

The logarithmic equation becomes a quadratic-style power equation.

Square roots

x - 1 = \pm 4

Both signs must be considered.

Solutions

x = 5 or x = - 3

Both create the positive logarithm argument sixteen.

Keep both branches: the even power produces two valid candidates.

x - 1 = 4 or x - 1 = - 4

Relay 06

Take logarithms to solve an exponential equation

When the exponential side is not a familiar power, logarithms bring the exponent down.

Reverse direction

Use a calculator-ready form

5^{x + 1} = 18

(x + 1) \ln 5 = \ln 18

x = \frac{\ln 18}{\ln 5} - 1 \approx 0.796

Relay 07

Match equal logarithms only when bases agree

A one-to-one logarithm function gives equal arguments from equal outputs.

Argument match

Write the shared domain first

\log_{2} (x - 1) = \log_{2} (5 - x)

1 < x < 5

x - 1 = 5 - x, x = 3

Relay 08

Keep log-domain and exponent rules separate

Exponential expressions are positive, but algebraic logarithm arguments still need explicit restrictions.

Domain control

Two different checks

Exponential input

b^{f (x)} > 0

The input is automatically positive for a valid positive base.

Algebraic input

g (x) > 0

This inequality must be solved before accepting candidates.

Route each equation by the location of the variable

The best first move depends on whether the unknown is in an argument, exponent, coefficient, or both.

Decision map

Four common routes

Convert to exponential form when the unknown is inside one logarithm argument.

Use the power rule when an outside coefficient blocks the conversion.

Match arguments when equal logarithms have the same valid base.

Take logarithms when the unknown remains in an exponent.

Candidate screening is part of the solution

Algebra can produce values that fail the original logarithmic equation.

Final filter

Three checks

List every algebraic candidate, including both signs from even powers.

Reject any value that makes an original logarithm argument nonpositive.

Substitute surviving values into the original equation, not only a transformed equation.

Skills Covered and How to Approach

Coordinate form conversion, exponent laws, logarithm properties, and domain reasoning.

Practice plan

Skills Covered

Core abilities developed by this practice test.

Convert between forms Preserve base, exponent, and result positions.

Use inverse relationships Cancel matching logarithmic and exponential functions.

Apply exponent rules Solve linear and even-power equations completely.

Take logarithms Bring an unknown exponent down as a multiplier.

Screen candidates Enforce the original logarithm domain.

How to Approach

A five-relay routine.

Write restrictions Require every original log argument to be positive.

Locate the variable Identify whether it is in an argument or exponent.

Choose the form Convert or take logs to expose the unknown.

Solve completely Keep both signs when an even power is involved.

Verify originally Test domain and equality in the starting equation.

Common Mistakes

Most errors come from switching forms incorrectly or skipping the original domain.

Relay repairs

Moving the wrong quantity

Track base, exponent, and result by position.

Dropping parentheses

Keep the complete logarithm argument grouped.

Cancelling unlike bases

Inverse cancellation requires matching bases.

Missing the negative root

An even-power equation can have two candidates.

Taking a log of only one side

Apply the same operation to both sides.

Rounding too early

Keep calculator precision until the final answer.

Matching unequal-base logs

Equal arguments follow only from the same one-to-one log function.

Accepting every algebraic candidate

Check each value in the original logarithmic equation.

Final relay audit

Inspect restrictions, form conversion, algebra, candidate count, and original substitution.

Course complete

Both tracks agree

A correct solution tells the same story in logarithmic and exponential form and survives every original-domain check.

Restrict · Convert · Solve · Verify

Were all original logarithm arguments restricted? Each input must remain positive.

Did base, exponent, and result keep their roles? Check the form conversion before solving.

Were exponent rules applied completely? Include both signs when required.

Was full calculator precision preserved? Round only the requested final value.

Was every answer tested in the original equation? Reject any candidate that fails domain or equality.

Use this free 20-question practice test for high school Advanced Algebra review, ACT-style skill practice, placement preparation, or classroom practice. You can retake the test without creating an account. The examples in this review block are illustrative and are not copies of the test questions.