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Exponential and Logarithmic Functions Practice Problems

Exponential and logarithmic functions undo each other, and most mixed problems test whether you can move between the two. These twelve problems cover inverse functions, telling exponential data from logarithmic data, and applied models where one form answers 'how much' and the other answers 'how long'.

  1. 1.

    For , find .

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    , so . Equivalently: which input gives ? .

  2. 2.

    The data and is exponential or logarithmic? Answer "exponential" or "logarithmic".

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    The inputs multiply by 2 while the outputs add 1 — the signature of a logarithm. It is .

  3. 3.

    The data and is exponential or logarithmic? Answer "exponential" or "logarithmic".

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    The inputs add 1 while the outputs multiply by , so the data is exponential: .

  4. 4.

    What is the domain of ? Use interval notation.

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    The logarithm accepts only positive inputs, because it returns the exponent that produced them and every power of a positive base is positive. Domain — the range of .

  5. 5.

    Find the inverse of . Type logs as log(argument, base), for example log(x, 3).

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    Swap and solve: , so and .

  6. 6.

    Find the inverse of .

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    Swap and solve: , so and .

  7. 7.

    A culture follows , with in hours. Find .

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    . Knowing the time and wanting the amount is an exponential question.

  8. 8.

    For the same culture, , when does the population reach 20,000? Give to two decimal places.

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    , so , and hours. Knowing the amount and wanting the time is a logarithmic question.

  9. 9.

    Sound level is decibels. Find for a sound 1000 times the reference intensity .

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    dB.

  10. 10.

    pH is . Find the pH when .

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    , so .

  11. 11.

    For , find .

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    The inverse of is , so .

  12. 12.

    The graph of passes through . Find .

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    means , so and .
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MPhil in Mathematics, Founder of Mathovia
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Exponential and Logarithmic Functions Practice Problems

These mixed problems follow the exponential vs logarithmic functions lesson, which compares the two function families side by side.

The inverse pair

Domain
Range
Key point
Asymptote

Finding an inverse

Replace with , swap and , and solve for . If the variable ends up in an exponent, convert to a logarithm; if it ends up inside a logarithm, convert to an exponential.

Common mistakes to watch for

  • Treating logs as division. is the exponent that needs to reach .
  • Negative or zero inputs. Logarithms are only defined for positive inputs.
  • Confusing the two graphs. An exponential has a horizontal asymptote; a logarithm has a vertical one.
  • Misusing the inverse relationship. and .

Frequently Asked Questions

How are exponential and logarithmic functions related?+

They are inverses. y = bˣ and y = log_b x undo each other, their graphs are reflections across y = x, and the domain of one is the range of the other.

How do I find the inverse of an exponential function?+

Swap x and y, isolate the power, then convert to logarithmic form. For y = 3ˣ + 1 that gives 3ʸ = x − 1 and y = log₃(x − 1).

How do I tell exponential data from logarithmic data?+

Look at which variable is multiplied. If x adds while y multiplies, the data is exponential. If x multiplies while y adds, it is logarithmic.

When do I use the exponential form and when the logarithmic form?+

If you know the time and want the amount, evaluate the exponential. If you know the amount and want the time, take a logarithm.

How do I type a logarithm in an answer?+

Use log(argument, base), for example log(x − 1, 3) for log base 3 of (x − 1). The answer is checked by value, so any equivalent form is accepted.

More practice problems

Logarithm Practice Problems Exponential Growth and Decay Practice Problems Solving Exponential Equations Practice Problems