Exponential and Logarithmic Models

Develop compound growth, half-life, logarithmic time.

Lesson 4 of 43 parts Practice Flashcards

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Compound Growth

Objective: Explain compound growth, apply the rule, and justify relevant restrictions.

A quantity compounded at fixed intervals follows a repeated multiplier, with a rate and time unit stated. Compare model predictions to context and data. The factor applied during each time interval. Identify the stated domain before using the rule, preserve restrictions during algebraic simplification, and interpret the result in the units or graph features requested. Use the worked example to connect the symbolic rule with a concrete calculation.

Formulas

$y=A b^t$

Worked examples

Try itStart at 5 and multiply by 3 each period. Find the value after 7 periods.Show answer

Step 1: Use yₜ=5·3^t. Step 2: At t=7, compute 3^7=2187. Step 3: Multiply by 5 to get 10935.

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Compound Growth

Objective: Explain compound growth, apply the rule, and justify relevant restrictions.

Formulas

$y=A b^t$

Worked examples

Try itStart at 5 and multiply by 3 each period. Find the value after 7 periods.Show answer

Step 1: Use yₜ=5·3^t. Step 2: At t=7, compute 3^7=2187. Step 3: Multiply by 5 to get 10935.

Half-Life

Objective: Explain half-life, apply the rule, and justify relevant restrictions.

After n half-life periods, a remaining population is A₀(1/2)^n in the ideal model. It describes a population trend, not a scheduled time for each item. One quarter of the initial amount. Identify the stated domain before using the rule, preserve restrictions during algebraic simplification, and interpret the result in the units or graph features requested. Use the worked example to connect the symbolic rule with a concrete calculation.

Formulas

$y=A(1−r)^t$

Worked examples

Try itStart at 640 and halve for 7 periods. What remains?Show answer

Step 1: Use the factor (1/2)^7. Step 2: Divide 640 by 2^7=128. Step 3: The remaining amount is 5.

Logarithmic Time

Objective: Explain logarithmic time, apply the rule, and justify relevant restrictions.

If A·b^t reaches target T with A>0 and b>0,b≠1, solve t=log_b(T/A) when T/A>0; interpret units and limits. The real logarithm argument must be positive. Identify the stated domain before using the rule, preserve restrictions during algebraic simplification, and interpret the result in the units or graph features requested. Use the worked example to connect the symbolic rule with a concrete calculation.

Formulas

$t=log_b(T/A)$

Worked examples

Try itAn amount starts at 5 and multiplies by 3 each period. After how many periods does it reach 10935?Show answer

Step 1: Write 5·3^t=10935. Step 2: Divide by 5: 3^t=2187. Step 3: Take log base 3: t=7 periods.

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