Exponential Functions
A free College Algebra lesson from the “Logarithms” unit, with a worked example and practice problems including step-by-step solutions.
An exponential function uses a constant multiplier repeatedly, with the variable in the exponent. The form y = a(b)^x shows an initial value a and a growth or decay factor b. This matters in college algebra because exponential models describe compound interest, population change, depreciation, and many natural growth processes. When practicing, identify whether the situation uses a constant ratio rather than a constant difference. A common mistake is treating a percent increase as simple addition each step instead of multiplying by a growth factor such as 1.08.
What you'll learn
- Evaluate exponential functions
- Identify growth and decay
- Use percent-change models
Worked example
Problem. Evaluate f(x) = 3(2^x) when x = 4.
- Substitute 4.
- 2^4 = 16.
- 3 x 16 = 48.
Answer: 48
Lesson preview: this page shows 5 practice items. The interactive activity contains 28 practice items and 15 quiz items (typed response and multiple choice). Counts include repeated prompts and variants.
Practice problems
1. Evaluate 2^6.
Show solution
- 2^6 = 64.
Answer: 64
2. Evaluate 5(3^2).
Show solution
- 3^2 = 9, then multiply by 5.
Answer: 45
3. A value grows by 8%. What multiplier is used each period?
Show solution
- 1 + 0.08 = 1.08.
Answer: 1.08
4. Evaluate 3^4.
Show solution
- 3 x 3 x 3 x 3 = 81.
Answer: 81
5. In y = 200(1.12)^t, what percent growth occurs each period?
Show solution
- 1.12 means 1 + 0.12, or 12% growth.
Answer: 12
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