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Solving Exponential Equations Practice Problems

An exponential equation hides the unknown in an exponent. The first four problems can be solved by rewriting both sides with the same base; the rest need logarithms, where you isolate the power, take ln of both sides and let the power rule bring the variable down.

  1. 1.

    Solve .

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    , so and .

  2. 2.

    Solve .

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    Write both as powers of 3: , so and .

  3. 3.

    Solve .

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    and , so and .

  4. 4.

    Solve .

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    , so and .

  5. 5.

    Solve to four decimal places.

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    Take of both sides: , so .

  6. 6.

    Solve to four decimal places.

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    .

  7. 7.

    Solve to four decimal places.

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    Isolate the power first: . Then , so .

  8. 8.

    Solve to four decimal places.

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    , so and .

  9. 9.

    Solve . Give both solutions, separated by a comma.

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    Same base, so , i.e. , which factors as . Both and are valid.

  10. 10.

    Solve to four decimal places.

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    . Collect: , so .

  11. 11.

    Solve to four decimal places.

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    Let : . is impossible, so and .

  12. 12.

    An account grows at 3% per year. How many years does it take to double? Give the answer to two decimal places.

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    The starting amount cancels: . So years.
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Solving Exponential Equations Practice Problems

Both methods used here are explained step by step in the solving exponential equations lesson.

Which method?

SituationMethod
both sides are powers of one numberrewrite with a common base, set exponents equal
no common baseisolate the power, take of both sides
unknown in exponents on both sidestake , distribute, collect the terms
and togethersubstitute and factor

The key step

Check each answer by substituting it back into the original equation, and discard any branch that would need a power of a positive base to be zero or negative.

Common mistakes to watch for

  • Taking logs too early. If both sides can be written with the same base, set the exponents equal.
  • Logging only part of a side. Isolate the exponential first: becomes .
  • Bringing down the wrong thing. , not .
  • Rounding too early. Keep exact logs until the last step.

Frequently Asked Questions

When can I solve an exponential equation without logarithms?+

When both sides can be written as powers of the same base. Then the exponents must be equal: 9ˣ = 27 becomes 3²ˣ = 3³, so x = 3/2.

What are the steps for solving with logarithms?+

Isolate the exponential term, take ln (or log) of both sides, use the power rule to bring the exponent down, and solve the linear equation that is left.

Why must I isolate the power before taking logs?+

Because there is no rule for the log of a sum. ln(4e^(0.5x) + 1) cannot be simplified, so divide and subtract until the exponential term stands alone.

Can an exponential equation have no solution?+

Yes. A positive base to any real power is positive, so an equation that reduces to eˣ = −3 has no real solution. In quadratic-form problems, reject any factor that gives a negative value for the exponential.

How accurate does my answer need to be?+

Round to the number of decimal places the question asks for. Answers within that rounding are accepted, so 2.7268 and 2.727 both pass for 3ˣ = 20.

More practice problems

Change of Base Practice Problems Solving Logarithmic Equations Practice Problems Exponential Growth and Decay Practice Problems