Exponential Function Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Exponential Function.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

An exponential function has the form f(x) = a \cdot b^x where b > 0, b \neq 1. The variable is in the exponent, not the base.

Growth (or decay) that multiplies by a constant factor repeatedly.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Exponential growth is eventually faster than any polynomial growth.

Common stuck point: 2^x grows much faster than x^2. By x = 10: 2^{10} = 1024, but 10^2 = 100.

Sense of Study hint: Make a table of values for x = 0, 1, 2, 3, 4 and watch how the outputs multiply by the same factor each step.

Worked Examples

Example 1

easy
A bacteria population starts at 500 and doubles every 3 hours. Write an exponential model and find the population after 9 hours.

Solution

  1. 1
    The general form is P(t) = P_0 \cdot b^{t/k} where P_0 = 500, b = 2, and k = 3.
  2. 2
    Model: P(t) = 500 \cdot 2^{t/3}.
  3. 3
    At t = 9: P(9) = 500 \cdot 2^{9/3} = 500 \cdot 2^3 = 500 \cdot 8 = 4000.

Answer

P(9) = 4000
Exponential growth models use the form P_0 \cdot b^{t/k} where b > 1 represents growth. The ratio t/k counts how many doubling periods have elapsed.

Example 2

medium
Solve 3^{2x - 1} = 81.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

medium
A radioactive substance has a half-life of 10 years. If the initial amount is 200 g, how much remains after 30 years?

Example 2

hard
Two bacteria colonies start with 100 and 500 cells respectively. Colony A doubles every 3 hours; Colony B doubles every 8 hours. After how many hours will Colony A surpass Colony B?
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Background Knowledge

These ideas may be useful before you work through the harder examples.

exponentsfunction definition