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Unit 6 Review and Quiz

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

This checkpoint confirms exponential and logarithmic fluency before discrete models. 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. A population starts at 30 and multiplies by 3 each year. Write the growth model A(x).

  1. Worked Example: First identify exactly what the question is asking: A population starts at 30 and multiplies by 3 each year. Write the growth model A(x).
  2. For exponential situations, identify the starting value and the repeated multiplier before calculating.
  3. Use A(x) = (starting value)(multiplier^x).
  4. The starting value is 30 and the multiplier is 3.
  5. So A(x) = 30(3^x).
  6. Check the result by substituting or estimating: the response should match 30(3^x) and make sense in the original problem.

Answer: 30(3^x)

Practice problems

1. Unit review 1 (Exponential Growth): A population starts at 30 and multiplies by 3 each year. Write the growth model A(x).

Show solution
  1. Unit Review: First identify exactly what the question is asking: A population starts at 30 and multiplies by 3 each year. Write the growth model A(x).
  2. For exponential situations, identify the starting value and the repeated multiplier before calculating.
  3. Use A(x) = (starting value)(multiplier^x).
  4. The starting value is 30 and the multiplier is 3.
  5. So A(x) = 30(3^x).
  6. Check the result by substituting or estimating: the response should match 30(3^x) and make sense in the original problem.

Answer: 30(3^x)

2. Unit review 2 (Exponential Decay): For A(x) = 48*(1/2)^x, find A(4).

Show solution
  1. Unit Review: First identify exactly what the question is asking: For A(x) = 48*(1/2)^x, find A(4).
  2. For fractions, use equivalent forms, common denominators, or reciprocals depending on the operation being used.
  3. Substitute x = 4: A(4) = 48*(1/2)^4.
  4. (1/2)^4 = 1/16, so divide 48 by 16.
  5. 48/16 = 3.
  6. Check the result by substituting or estimating: the response should match 3 and make sense in the original problem.

Answer: 3

3. Unit review 3 (Transformations of Exponential Functions): The parent function y = 3^x has horizontal asymptote y = c. Find c.

Show solution
  1. As x decreases, 3^x approaches 0 but never reaches it.
  2. The curve hugs the x-axis on the left.
  3. So the asymptote is y = 0, giving c = 0.

Answer: 0

4. Unit review 4 (Logarithms as Inverses): A log of its own base: find log base 2 of 2.

Show solution
  1. Unit Review: First identify exactly what the question is asking: A log of its own base: find log base 2 of 2.
  2. For logarithms, rewrite the statement as an exponent question so the base, exponent, and result are clear.
  3. Ask: 2 to what power gives 2?
  4. 2^1 = 2.
  5. So log base 2 of 2 = 1.
  6. Check the result by substituting or estimating: the response should match 1 and make sense in the original problem.

Answer: 1

5. Unit review 5 (Logarithm Rules): Expand log(xy) using a log rule.

Choices: log(x) + log(y) · log(x) - log(y) · log(x)log(y) · log(x + y)

Show solution
  1. The argument xy is a product.
  2. Apply the product rule: log(xy) = log(x) + log(y).
  3. A product inside becomes a sum outside.

Answer: log(x) + log(y)

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