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Solving Exponential and Logarithmic Equations

A free Precalculus lesson from the “Exponential and Logarithmic Functions” unit, with a worked example and practice problems including step-by-step solutions.

Solving exponential and log equations means choosing the inverse form and checking any domain restrictions. This lesson is part of Precalculus: Advanced Functions, so the emphasis is on interpreting behavior, choosing the right representation, and explaining the result clearly rather than memorizing isolated algebra moves.

What you'll learn

Why it matters: Exponential and logarithmic models describe growth, decay, sound, pH, finance, and scientific scales.

Worked example

Problem. Solve 3^x = 3^3.

  1. Worked Example: First identify exactly what the question is asking: Solve 3^x = 3^3.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Both sides are powers of 3, so the bases already match.
  4. Equal bases force equal exponents, so x = 3.
  5. So x = 3.
  6. Check the result by substituting or estimating: the response should match 3 and make sense in the original problem.

Answer: 3

Practice problems

1. Solve 3^x = 3^3.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Solve 3^x = 3^3.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Both sides are powers of 3, so the bases already match.
  4. Equal bases force equal exponents, so x = 3.
  5. So x = 3.
  6. Check the result by substituting or estimating: the response should match 3 and make sense in the original problem.

Answer: 3

2. Solve 2^x = 16.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Solve 2^x = 16.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Write 16 as a power of 2: 2^4 = 16.
  4. Now 2^x = 2^4, so the exponents match.
  5. So x = 4.
  6. Check the result by substituting or estimating: the response should match 4 and make sense in the original problem.

Answer: 4

3. Solve log base 3 of x = 2.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Solve log base 3 of x = 2.
  2. For logarithms, rewrite the statement as an exponent question so the base, exponent, and result are clear.
  3. Rewrite in exponential form: 3^2 = x.
  4. 3^2 = 9.
  5. So x = 9, and 9 > 0 so it is a valid input.
  6. Check the result by substituting or estimating: the response should match 9 and make sense in the original problem.

Answer: 9

4. Solve 2^x = 1.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Solve 2^x = 1.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Any nonzero base raised to the 0 power equals 1.
  4. So 2^x = 1 forces x = 0.
  5. So x = 0.
  6. Check the result by substituting or estimating: the response should match 0 and make sense in the original problem.

Answer: 0

5. Solve 3^x = 3.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Solve 3^x = 3.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Write the right side as 3^1.
  4. Then 3^x = 3^1, so the exponents match.
  5. So x = 1.
  6. Check the result by substituting or estimating: the response should match 1 and make sense in the original problem.

Answer: 1

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