2.3: Rescaling the Exponent | AP Precalculus

How it works

An exponent counts how many growth periods have elapsed. If a quantity doubles every d time units, use time divided by d. Equivalent models can express the same growth with different time units or bases.

P(t)=P_0b^{t/d}=P_0\left(b^{1/d}\right)^t

Worked examples

Example 1

A quantity doubles every $\displaystyle 3$ hours from an initial $\displaystyle 10$ . Give a model and its value after $\displaystyle 6$ hours.

$\displaystyle P(t)=10\cdot2^{t/3}$ ; $\displaystyle P(6)=40$ .

Example 2

A substance halves every $\displaystyle 5$ days from an initial $\displaystyle 80$ g. Find the mass after $\displaystyle 10$ days.

$\displaystyle M(t)=80(1/2)^{t/5}$ , so $\displaystyle M(10)=20$ g.

Problems

Problem 2.3.1

A quantity starts at $\displaystyle 12$ and triples every $\displaystyle 2$ days. Find its value after $\displaystyle 4$ days.

Show worked solution

$\displaystyle P(t)=12\cdot3^{t/2}$ , so $\displaystyle P(4)=108$ .

Problem 2.3.2

Rewrite $\displaystyle 100\cdot4^{t/2}$ using a per-unit base.

Show worked solution

Because $\displaystyle 4^{1/2}=2$ , the equivalent model is $\displaystyle 100\cdot2^t$ .

Problem 2.3.3

A model uses $\displaystyle t$ in years: $\displaystyle P(t)=50(1.08)^t$ . Rewrite using months m.

Show worked solution

Substitute $\displaystyle t=m/12$ : $\displaystyle P(m)=50(1.08)^{m/12}$ . The monthly factor is $\displaystyle (1.08)^{1/12}$ .

Lesson Problems PDF Solutions PDF Unit MCQ

How it works

Worked examples

Example 1

Example 2

Problems

Problem 2.3.1

Problem 2.3.2

Problem 2.3.3

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