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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 quantity starts at 100 and grows by a factor of 3 each period. Write A(t).

  1. Worked Example: First identify exactly what the question is asking: A quantity starts at 100 and grows by a factor of 3 each period. Write A(t).
  2. Use the structure of the expression to choose a factoring pattern, then check that the factors multiply back to the original expression.
  3. Use starting value times multiplier to the t power.
  4. The starting value is 100.
  5. The multiplier is 3.
  6. Check the result by substituting or estimating: the response should match 100(3^t) and make sense in the original problem.

Answer: 100(3^t)

Practice problems

1. Unit review 1 (Exponential Growth): A quantity starts at 100 and grows by a factor of 3 each period. Write A(t).

Show solution
  1. Unit Review: First identify exactly what the question is asking: A quantity starts at 100 and grows by a factor of 3 each period. Write A(t).
  2. Use the structure of the expression to choose a factoring pattern, then check that the factors multiply back to the original expression.
  3. Use starting value times multiplier to the t power.
  4. The starting value is 100.
  5. The multiplier is 3.
  6. Check the result by substituting or estimating: the response should match 100(3^t) and make sense in the original problem.

Answer: 100(3^t)

2. Unit review 2 (Exponential Decay): A quantity starts at 160 and decreases by 20 percent each period. What multiplier is used?

Show solution
  1. Unit Review: First identify exactly what the question is asking: A quantity starts at 160 and decreases by 20 percent each period. What multiplier is used?
  2. For percents, convert the percent to a decimal or fraction and connect it to the base amount in the problem.
  3. A 20 percent decrease leaves 80 percent.
  4. 80 percent as a decimal is 0.8.
  5. That is the repeated multiplier.
  6. Check the result by substituting or estimating: the response should match 0.8 and make sense in the original problem.

Answer: 0.8

3. Unit review 3 (Transformations of Exponential Functions): Compared with y = 2^x, y = 2^x + 5 has horizontal asymptote:

Choices: y = 5 · y = 0 · x = 5 · y = -5

Show solution
  1. Unit Review: First identify exactly what the question is asking: Compared with y = 2^x, y = 2^x + 5 has horizontal asymptote:
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. The parent exponential has asymptote y = 0.
  4. Adding 5 shifts the graph up 5.
  5. The asymptote becomes y = 5.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: y = 5

4. Unit review 4 (Logarithms as Inverses): Which exponential form matches log base 3 of 9 = 2?

Choices: 3^2 = 9 · 9^3 = 2 · 3^9 = 2 · 2^3 = 9

Show solution
  1. Unit Review: First identify exactly what the question is asking: Which exponential form matches log base 3 of 9 = 2?
  2. For logarithms, rewrite the statement as an exponent question so the base, exponent, and result are clear.
  3. A logarithm is an exponent question.
  4. The base stays 3.
  5. The exponent is 2 and the result is 9.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: 3^2 = 9

5. Unit review 5 (Logarithm Rules): A common log-rule mistake is:

Choices: splitting log(a + b) into log(a) + log(b) · using log(ab) = log(a) + log(b) · moving an exponent down as a coefficient · rewriting a quotient as a difference

Show solution
  1. Unit Review: First identify exactly what the question is asking: A common log-rule mistake is:
  2. For logarithms, rewrite the statement as an exponent question so the base, exponent, and result are clear.
  3. There is no sum rule for logs.
  4. Product, quotient, and power rules are valid.
  5. Addition inside a log must stay together.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: splitting log(a + b) into log(a) + log(b)

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