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Solving Logarithmic Equations Practice Problems

Solving a log equation means getting rid of the logarithm: convert to exponential form, or match two logs with the same base and drop them. Several of these twelve problems produce a second, extraneous answer — enter only the solutions that keep every argument positive.

  1. 1.

    Solve .

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    Exponential form: , so . The argument is positive.

  2. 2.

    Solve .

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    , so and .

  3. 3.

    Solve .

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    Same base on both sides, so the arguments are equal: , giving . Both arguments equal , which is positive.

  4. 4.

    Solve to four decimal places.

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    The base of is : , so .

  5. 5.

    Solve .

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    Quotient rule: , so . Then and .

  6. 6.

    Solve .

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    Condense: , so . Then and . Check: and .

  7. 7.

    Solve .

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    Condense the left side first: . Then and . Dropping the logs before condensing would wrongly give .

  8. 8.

    Solve .

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    Power rule: , so and . But is undefined in the original equation, so only is a solution.

  9. 9.

    Find the true solution of .

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    Condense: , so . makes undefined, so it is extraneous. The true solution is .

  10. 10.

    Solve .

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    , so . Reject ; the solution is .

  11. 11.

    Solve .

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    , so . is extraneous; .

  12. 12.

    Solve .

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    , so . would put a negative number inside , so the only solution is .
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MPhil in Mathematics, Founder of Mathovia
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Solving Logarithmic Equations Practice Problems

The four-step method below is worked through in the solving logarithmic equations lesson.

The two tools

The four steps

StepAction
1Condense each side to a single logarithm
2Remove the log: exponential form or equal arguments
3Solve the resulting equation
4Check every candidate in the original equation

Why step 4 matters

Condensing into enlarges the domain: the original needs , but the condensed form also accepts . Any root that lands in that extra region is extraneous.

Common mistakes to watch for

  • Skipping the check. Solutions that make any log argument zero or negative must be rejected.
  • Combining logs incorrectly. , not .
  • Not exponentiating correctly. means .
  • Dropping a log from one side. Condense to one log on each side before setting the arguments equal.

Frequently Asked Questions

How do I get rid of a logarithm in an equation?+

If a log equals a number, convert to exponential form: log_b M = c becomes M = b^c. If a log equals a log with the same base, drop both: M = N.

What is an extraneous solution?+

A value that solves the simplified equation but makes the argument of a logarithm zero or negative in the original. It usually appears after condensing several logs into one, and must be rejected.

Should I reject every negative answer?+

No. Reject an answer only if it makes some argument non-positive. In ln(x + 4) = 0 the answer x = −3 is valid, because the argument becomes 1.

Why condense before removing the logs?+

Because the one-to-one property only applies to a single log on each side. log x + log 3 = log 12 must become log 3x = log 12 first, giving x = 4, not x = 9.

What happens if I enter both roots of the quadratic?+

It is marked wrong when one of them is extraneous. Check each candidate in the original equation and enter only the ones that keep every argument positive.

More practice problems

Solving Exponential Equations Practice Problems Logarithm Rules Practice Problems Common and Natural Log Practice Problems