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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

Why it matters: Compound interest, depreciation, population growth, medicine decay, and viral spread all use repeated multiplication. The base is the multiplier, so values above 1 grow and values between 0 and 1 decay.

Worked example

Problem. Evaluate f(x) = 3(2^x) when x = 4.

  1. Substitute 4.
  2. 2^4 = 16.
  3. 3 x 16 = 48.

Answer: 48

Practice problems

1. Evaluate 2^6.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Evaluate 2^6.
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. 2^6 = 64.
  4. Check the result by substituting or estimating: the response should match 64 and make sense in the original problem.

Answer: 64

2. Evaluate 5(3^2).

Show solution
  1. Core Practice: First identify exactly what the question is asking: Evaluate 5(3^2).
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. 3^2 = 9, then multiply by 5.
  4. Check the result by substituting or estimating: the response should match 45 and make sense in the original problem.

Answer: 45

3. A value grows by 8%. What multiplier is used each period?

Show solution
  1. Challenge: First identify exactly what the question is asking: A value grows by 8%. What multiplier is used each period?
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. 1 + 0.08 = 1.08.
  4. Check the result by substituting or estimating: the response should match 1.08 and make sense in the original problem.

Answer: 1.08

4. Evaluate 3^4.

Show solution
  1. Evaluation: First identify exactly what the question is asking: Evaluate 3^4.
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. 3 x 3 x 3 x 3 = 81.
  4. Check the result by substituting or estimating: the response should match 81 and make sense in the original problem.

Answer: 81

5. In y = 200(1.12)^t, what percent growth occurs each period?

Show solution
  1. Growth: First identify exactly what the question is asking: In y = 200(1.12)^t, what percent growth occurs each period?
  2. For exponential situations, identify the starting value and the repeated multiplier before calculating.
  3. 1.12 means 1 + 0.12, or 12% growth.
  4. Check the result by substituting or estimating: the response should match 12 and make sense in the original problem.

Answer: 12

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