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Change of Base Practice Problems

The change of base formula rewrites a logarithm in any base as a ratio your calculator can handle: the argument's log on top, the base's log underneath. These twelve problems use it for decimal answers, then for exact fractions by switching to a friendlier base.

  1. 1.

    Evaluate to four decimal places.

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    . Sense check: and , so the answer is a little above 3.

  2. 2.

    Evaluate to four decimal places.

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    .

  3. 3.

    Evaluate .

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    exactly, because .

  4. 4.

    Evaluate to four decimal places.

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    Using common logs: .

  5. 5.

    Evaluate to four decimal places.

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    .

  6. 6.

    Evaluate to four decimal places.

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    . The argument is below 1 and the base above 1, so the answer is negative.

  7. 7.

    Find the exact value of .

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    Change to base 2: .

  8. 8.

    Find the exact value of .

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    Change to base 3: .

  9. 9.

    Find the exact value of .

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    Change to base 5: .

  10. 10.

    Evaluate .

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    Write both with natural logs: . The cancels.

  11. 11.

    Solve to four decimal places.

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    Logarithmic form: .

  12. 12.

    Evaluate to four decimal places.

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    . A base below 1 turns an argument above 1 into a negative log.
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MPhil in Mathematics, Founder of Mathovia
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Change of Base Practice Problems

The formula and its proof are in the change of base formula lesson. Keep six or more decimal places in the intermediate logs and round only at the end.

The formula

Choosing a new base

GoalNew base
a decimal answer on a calculator or
an exact answera number that both and are powers of
a product of logs in different basesany common base — the middle terms cancel

Two identities that follow

Common mistakes to watch for

  • Flipping the fraction. : the base goes on the bottom.
  • Mixing log types. Use the same log (common or natural) on top and bottom.
  • Rounding too early. Round only the final answer.
  • Forgetting to sanity-check. must be between 3 and 4, since and .

Frequently Asked Questions

What is the change of base formula?+

log_b x = log x ÷ log b = ln x ÷ ln b. The argument's log goes on top and the old base's log goes underneath. Any new base works, as long as the same one is used top and bottom.

Should I use log or ln?+

Either — the ratio is identical. log 200 ÷ log 6 and ln 200 ÷ ln 6 both give about 2.9570. Just do not mix them in one fraction.

What is the most common mistake?+

Putting the base on top. ln 2 ÷ ln 10 is log₁₀ 2 ≈ 0.3010, the reciprocal of log₂ 10 ≈ 3.3219. Argument on top, base on the bottom.

When can I get an exact answer instead of a decimal?+

When the base and the argument are powers of the same number. Change to that number as the new base: log₈ 32 = log₂ 32 ÷ log₂ 8 = 5/3.

How do I sanity-check a decimal answer?+

Find the two whole-number powers of the base that bracket the argument. log₂ 10 must sit between 3 and 4, because 2³ = 8 and 2⁴ = 16.

More practice problems

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