Skip to main content
Mathovia

Math Practice

Exponential vs Logistic Growth Practice Problems

Start by deciding whether anything limits the growth, then read the carrying capacity and starting value straight off the logistic formula. The harder problems ask when the population reaches a target, which means isolating e^(−kt) and taking a natural log.

  1. 1.

    Fish are stocked in a pond with a limited food supply. Is their population better modeled by exponential or logistic growth? Answer "exponential" or "logistic".

    Show full solution

    The food supply sets a ceiling, the carrying capacity, so growth slows as the pond fills up. That is logistic growth.

  2. 2.

    A savings balance earns a fixed 3% a year with nothing withdrawn. Exponential or logistic? Answer "exponential" or "logistic".

    Show full solution

    Nothing limits the balance and the rate stays the same, so it grows exponentially.

  3. 3.

    A population follows . What is the carrying capacity?

    Show full solution

    In the numerator is the carrying capacity, so .

  4. 4.

    For the same model, , what is the starting population ?

    Show full solution

    At , , so .

  5. 5.

    A logistic population has carrying capacity 1,000 and starts at 50. What is the constant in ?

    Show full solution

    .

  6. 6.

    Using , find the population at , to the nearest whole number.

    Show full solution

    , so about 411.

  7. 7.

    A logistic population has carrying capacity 3,600. At what population size is it growing fastest?

    Show full solution

    Logistic growth is fastest at the inflection point, half the carrying capacity: .

  8. 8.

    A population of 200 grows exponentially with . What is the population at , to the nearest whole number?

    Show full solution

    .

  9. 9.

    For , at what time is the population growing fastest? Give two decimal places.

    Show full solution

    The inflection point is at , when .

  10. 10.

    A logistic population has , and . What is the population at , to the nearest whole number?

    Show full solution

    , so .

  11. 11.

    For the same population (, , ), when does it reach 4,000? Give two decimal places.

    Show full solution

    , so and .

  12. 12.

    A population of 600 lives in a habitat with carrying capacity 800. Its logistic growth rate is what percentage of the exponential rate at the same size?

    Show full solution

    The logistic rate is . The extra factor is , so growth runs at 25% of the exponential rate.
Written by
MPhil in Mathematics, Founder of Mathovia
Published

Mathematics verification: Mathematical formulas, calculations, worked examples and solutions are reviewed for accuracy before publication. How we check our work

Exponential vs Logistic Growth Practice Problems

These problems follow the exponential vs logistic growth lesson. The time questions use the same steps as solving exponential equations.

The two models

FormulaKey facts
Exponentialno upper limit
Logistic, levels off at
Fastest logistic growthat

Before you calculate

Check the sign in the exponent: a logistic model uses . When solving for time, isolate the exponential term completely before taking the natural log.

Common mistakes to watch for

  • Expecting logistic growth to keep accelerating. Logistic growth slows as it nears the carrying capacity.
  • Mixing up the carrying capacity. In , is the maximum population.
  • Using the exponential model too long. Exponential growth fits early data but overshoots once resources limit growth.
  • Misplacing the inflection point. Logistic growth is fastest at half the carrying capacity.

Frequently Asked Questions

How do I tell whether a problem is exponential or logistic?+

Look for a limit. If the description mentions limited food, space, customers or people, or says growth slows as the population gets bigger, it is logistic. If nothing caps the growth, it is exponential.

How do I read K and the starting value from a logistic formula?+

In P(t) = K / (1 + Ae^(−kt)) the numerator is the carrying capacity K. The starting value is K / (1 + A), found by setting t = 0.

How do I solve for the time in a logistic model?+

Set the formula equal to the target, take reciprocals to isolate 1 + Ae^(−kt), subtract 1, divide by A, then take the natural log of both sides and divide by −k.

How should I enter my answers?+

Type numbers without commas, for example 1800. Round to the place the question asks for; answers within the stated rounding are marked correct. For the classification questions, type exponential or logistic.

More practice problems

Exponential Growth and Decay Practice Problems Solving Exponential Equations Practice Problems Continuous Compound Interest Practice Problems