Exponents and Logarithms Practice Test

Connect exponential and logarithmic forms, apply logarithm rules, and solve equations.

Exponents and Logarithms Practice Test

20 mixed exponent and logarithm questions with worked solutions.

Instant feedback · Worked explanations

Inverse Function Observatory

Read the power one way. Read the logarithm backward.

Exponentials and logarithms are inverse descriptions of the same relationship. One form asks for the result of repeated multiplication; the other asks for the exponent that produces a given result. Once that inverse link is secure, exact evaluations, logarithm rules, change of base, domains, and equations become parts of one coherent system rather than separate tricks.

ConvertIdentify base, exponent, and result.

EvaluateAsk which exponent produces the argument.

RewriteUse product, quotient, and power rules only on valid structures.

SolveIsolate the exponential or logarithm before applying its inverse.

Check domainEvery logarithm argument must be positive.

1. Exponential and logarithmic forms are inverse statements

The base stays the same. The exponent in exponential form becomes the logarithm value in logarithmic form.

Exponential reading

b^{x} = y

The base is raised to an exponent to produce a result.

inverse pair⇄

Logarithmic reading

\log_{b} (y) = x

The logarithm asks which exponent on the base produces the argument.

2. The graphs reveal the inverse relationship visually

For a base greater than one, the exponential graph rises rapidly while the logarithmic graph rises slowly. They reflect across the diagonal line that represents equal input and output.

Inverse pair

exponential logarithm inverse-reflection line

horizontal inputvertical output

The two curves exchange the roles of input and output. That is why converting between exponential and logarithmic notation reverses which quantity is treated as the exponent.

Exponential growth

A base greater than one produces increasing growth.

Exponential decay

A base between zero and one produces decreasing decay.

3. Exact logarithm values come from recognizing powers

Before reaching for a calculator, rewrite the argument as a power of the base when possible.

Exact value

Recognize a familiar power

\log_{2} (32) = 5

The logarithm value is the exponent required to produce the argument.

Natural logarithm

Base of the natural exponential

\ln (e^{x}) = x

Natural logarithms reverse exponential expressions with the natural base.

Inverse thinking

Ask the exponent question

Instead of treating a logarithm like division, identify the base and ask which power produces the argument.

4. Product, quotient, and power rules translate multiplication structure

Logarithm rules work because exponents turn multiplication into addition and division into subtraction.

Product rule

Multiplication becomes addition

\log_{b} (a \cdot c) = \log_{b} (a) + \log_{b} (c)

Use this rule only when the logarithm contains a product.

Quotient rule

Division becomes subtraction

\log_{b} (\frac{a}{c}) = \log_{b} (a) - \log_{b} (c)

Keep numerator and denominator order consistent.

Power rule

Exponent becomes a coefficient

\log_{b} (a^{x}) = x \log_{b} (a)

The exponent moves in front of the logarithm as a multiplier.

\log_{b} (a + c) \neq \log_{b} (a) + \log_{b} (c)

Addition inside a logarithm does not split into a sum of logarithms. The rules correspond to multiplication, division, and powers—not ordinary addition.

5. Expanding and condensing are opposite rewriting directions

Expansion separates a logarithm into a sum or difference. Condensing rebuilds a single logarithm from coefficients and operations.

Expand

\log_{b} (\frac{x^{2}}{y})

2 \log_{b} (x) - \log_{b} (y)

Use the power rule and quotient rule in reverse order.

⇄

Condense

2 \log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x^{2}}{y})

Move coefficients back into exponents, then combine with product or quotient structure.

6. Change of base rewrites an unfamiliar logarithm using a familiar one

The numerator is the logarithm of the argument; the denominator is the logarithm of the original base.

Natural logs

Change to natural logarithms

\log_{b} (x) = \frac{\ln (x)}{\ln (b)}

This form is convenient when a calculator provides natural logarithms.

Common logs

Change to common logarithms

\log_{b} (x) = \frac{\log (x)}{\log (b)}

The same ratio works with any valid new base.

Common mistake

Do not reverse the fraction

Argument goes on top; original base goes on the bottom.

7. Equation strategy depends on whether a common base is available

Isolate the exponential or logarithmic expression first. Then apply the inverse relationship.

Matching-base exponentialRewrite both sides with the same base.

Equation

3^{x} = 81 \Rightarrow x = 4

Why it works

Equal powers of the same valid base have equal exponents.

Nonmatching-base exponentialUse logarithms after isolating the exponential term.

Equation

5^{x} = 17 \Rightarrow x = \frac{\ln (17)}{\ln (5)}

Why it works

A logarithm brings the exponent down so the variable can be isolated.

Logarithmic equationRewrite in exponential form.

Equation

\log_{2} (x - 1) = 3 \Rightarrow x - 1 = 8 \Rightarrow x = 9

Final check

The logarithm argument must remain positive after substitution.

8. Logarithm domains are strict: every argument must be positive

A logarithm of zero or a negative real number is not defined in the real-number system.

Domain gate

x - 4 > 0 \Rightarrow x > 4

Check the complete argument of every logarithm before accepting a solution.

Argument positivityEvery logarithm input must be strictly greater than zero.

Candidate screeningA transformed equation may produce a candidate that makes an original logarithm invalid.

Base restrictionsA logarithm base must be positive and cannot equal one.

9. Growth and decay models connect exponents to real change over time

Exponential models describe repeated proportional change, while logarithms solve backward for time or another exponent.

Growth model

P (t) = P_{0} {(1 + r)}^{t}

A repeated percentage increase creates an exponential factor greater than one.

⇄

Decay model

A (t) = A_{0} {(1 - r)}^{t}

A repeated percentage decrease creates a factor between zero and one.

10. Error analysis: most mistakes come from using a valid rule on the wrong structure

Before applying a logarithm identity, inspect the operation inside the logarithm and confirm the domain.

Logarithm distributed over addition

$\log_{b} (a + c) \neq \log_{b} (a) + \log_{b} (c)$

Change-of-base fraction reversed

Use logarithm of the argument over logarithm of the original base.

Domain restriction ignored

A logarithmic equation can produce algebraic candidates that make an original argument nonpositive.

Logarithm treated like ordinary division

A logarithm is an exponent question, not a fraction.

Power rule used on addition

The power rule moves an actual exponent, not an additive term.

Exponential not isolated before logging

Separate the exponential expression before applying a logarithm whenever possible.

Final exponent-log audit

Before accepting an answer, verify the inverse relationship, rule structure, domain, and final equation check.

Did I identify the base, exponent, and result correctly?Keep these roles fixed when converting between forms.

Did I apply product, quotient, or power rules only to matching structures?Do not distribute logarithms over addition.

Did I isolate the exponential or logarithm before using its inverse?This keeps equation solving controlled and readable.

Did I use change of base in the correct order?Argument on top, original base on the bottom.

Are all logarithm arguments positive?Reject any candidate that violates an original logarithm domain.

Did I keep exact form when the problem permits it?Use decimal approximation only when needed.