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Logarithm Practice Problems

A logarithm asks one question: what power of the base gives this number? Every problem here is that question in some form. Start by converting each log to exponential form, and the harder items — fractional answers, bases below 1, unknown bases — work out the same way.

  1. 1.

    Evaluate .

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    Ask . Since , .

  2. 2.

    Evaluate .

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    , so .

  3. 3.

    Evaluate .

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    Any base to the power 0 is 1, so for every valid base.

  4. 4.

    Evaluate .

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    , so . The log of its own base is always 1.

  5. 5.

    Evaluate .

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    , so . A negative log is fine; only a negative argument is not.

  6. 6.

    Evaluate .

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    , so .

  7. 7.

    Evaluate .

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    Write both as powers of 2: becomes , so and .

  8. 8.

    Solve .

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    Exponential form: , so . A logarithm's base must be positive, so .

  9. 9.

    Evaluate .

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    becomes , so and .

  10. 10.

    Solve .

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    Exponential form: , so and . Check: the argument is positive.

  11. 11.

    Evaluate .

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    becomes , so .

  12. 12.

    Solve .

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    Exponential form: .
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MPhil in Mathematics, Founder of Mathovia
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Logarithm Practice Problems

Work through the logarithms lesson first if the definition is new. Every problem below starts with the same move: convert to exponential form.

The definition

Values worth knowing

PropertyStatement
log of 1
log of the base
log of a power of the base
reciprocal argument

When the numbers are not obvious

If neither the argument nor the base is a clean power of the other, write both as powers of a common number — usually 2 or 3 — and set the exponents equal. That is how fractional answers such as come out exactly.

Common mistakes to watch for

  • Reading backwards. It asks ” to what power gives ?”
  • Zero or negative inputs. and are undefined.
  • Forgetting special values. and for every base.
  • Assuming answers are whole numbers. is about .

Frequently Asked Questions

What does log_b x actually mean?+

It is the exponent that b must be raised to in order to give x. log₂ 8 = 3 says exactly the same thing as 2³ = 8, just solved for the exponent.

How do I evaluate a logarithm without a calculator?+

Rewrite the argument as a power of the base. If log₈ 4 is awkward, write 8 = 2³ and 4 = 2²; the question becomes 2^(3y) = 2², so y = 2/3.

Can a logarithm be negative?+

The answer can be negative — log₃(1/9) = −2 — because negative exponents give numbers between 0 and 1. The argument cannot be negative or zero, because no power of a positive base is.

Why must the base be positive and not 1?+

A negative base gives undefined values for many exponents, and base 1 gives 1 for every exponent, so it could never reach any other number. That is why the base is restricted to b > 0, b ≠ 1.

How do I solve an equation like log₃(2x − 1) = 2?+

Convert it to exponential form, 2x − 1 = 3², and solve the resulting equation. Then check that the argument is positive at your answer.

More practice problems

Common and Natural Log Practice Problems Logarithm Rules Practice Problems Change of Base Practice Problems