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Grade 11 · lesson 4 of 4

Recover a Period Count from a Growth Target

An exponential model can be solved by first dividing out the initial amount.

About 25 minutes, plus any extended practice

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Supplied materials

  • Paper or a browser scratch area; a calculator is optional unless rounding is requested.
  • Supplied teaching case: A model is A(t)=2·2^t. At what t does A(t)=16? Candidate answers: (A) 2 periods (B) 3 periods (C) 4 periods (D) 8 periods.
  • Independent check inputs: A model is A(t)=3·2^t. At what t does A(t)=24? Candidate answers: (A) 2 periods (B) 3 periods (C) 4 periods (D) 8 periods.

Learn the idea

Logarithms reverse exponentiation with a valid positive base other than one. They answer exponent questions, reorganize multiplicative structure and solve exponential relationships while requiring positive real logarithm arguments. An exponential model can be solved by first dividing out the initial amount. If the remaining ratio is an exact power of the base, equal exponents identify the period count directly.

Here is the method on the supplied teaching case. Divide by 2: 2^t=8=2³. Equal positive-base powers give t=3. A logarithm could solve the same equation when the target is not an exact power.

The period count describes the supplied model rather than guaranteed real growth. A target that is not an exact base power would need logarithms or approximation; adding ratios would describe a different process.

  1. Write or sketch the supplied teaching case, labeling its quantities and restrictions.
  2. Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
  3. Solve the independent check with its changed inputs before revealing feedback.
  4. Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.
Worked example

See the method in action

Teaching case: A model is A(t)=2·2^t. At what t does A(t)=16? Candidate answers: (A) 2 periods (B) 3 periods (C) 4 periods (D) 8 periods. Solution: Divide by 2: 2^t=8=2³. Equal positive-base powers give t=3. A logarithm could solve the same equation when the target is not an exact power. The final answer is 3 periods. The period count describes the supplied model rather than guaranteed real growth. A target that is not an exact base power would need logarithms or approximation; adding ratios would describe a different process.

Your turn

Check your reasoning

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Helpful tips
  • An exponential model can be solved by first dividing out the initial amount.
  • The period count describes the supplied model rather than guaranteed real growth.
A model is A(t)=3·2^t. At what t does A(t)=24?

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See the answer and explanation

3 periods

Divide by 3: 2^t=8=2³. Equal positive-base powers give t=3. A logarithm could solve the same equation when the target is not an exact power.

Take it further

Changed-input task: A model is A(t)=9·2^t. At what t does A(t)=72? Candidate answers: (A) 2 periods (B) 3 periods (C) 4 periods (D) 8 periods. Show the mathematical reason, not only the selected answer. Expected result: 3 periods. Divide by 9: 2^t=8=2³. Equal positive-base powers give t=3. A logarithm could solve the same equation when the target is not an exact power.

Nothing here is saved to an account or sent to a teacher. The answer download includes your current choice and any checked feedback. It does not include choices from the separate More practice section. Download that section separately.

More practice: try new cases

Choose an answer for each problem. Explain your method on paper, then use the feedback to decide what to revisit.

More practice: try new cases
1. A model is A(t)=4·2^t. At what t does A(t)=32?
Need a hint?
  • An exponential model can be solved by first dividing out the initial amount.
  • The period count describes the supplied model rather than guaranteed real growth.
2. A model is A(t)=5·2^t. At what t does A(t)=40?
Need a hint?
  • An exponential model can be solved by first dividing out the initial amount.
  • The period count describes the supplied model rather than guaranteed real growth.
3. A model is A(t)=6·2^t. At what t does A(t)=48?
Need a hint?
  • An exponential model can be solved by first dividing out the initial amount.
  • The period count describes the supplied model rather than guaranteed real growth.

Your choices are temporary. Feedback appears after you check all the answers.

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These LifeStarter passages, scenarios and exercises are authored for this Academy with AI assistance and review. External resources keep their publishers’ terms. A grade label suggests a starting level; it does not establish alignment with your school’s complete requirements.