Modeling with Exponentials and Logarithms

What You’ll Learn

In this lesson you’ll learn how to create and use exponential and logarithmic models for real-world growth, decay, and other phenomena. You saw exponential growth and decay in Algebra 2. Here we go deeper into modeling and solving for unknowns using logarithms.

The Concept

Exponential and logarithmic functions are powerful modeling tools because many natural processes involve constant percentage change.

Common models:

Exponential growth (population, investment, spread):

$$P(t) = P_0 (1 + r)^t$$

where $P_0$ is the initial amount, r is the growth rate per period, and t is time.

Exponential decay (radioactive decay, depreciation, cooling):

$$P(t) = P_0 (1 - r)^t \quad \text{or} \quad P(t) = P_0 \left(\frac{1}{2}\right)^{t/h}$$

where h is the half-life.

To solve for time or rate, use logarithms:

$$t = \frac{\log(P / P_0)}{\log(1 + r)}$$

Worked Example

Example 1: Bacterial growth

A population of 500 bacteria grows at 12% per hour. Find the population after 8 hours.

The model:

$$P(t) = 500(1.12)^t$$

After 8 hours:

$$P(8) = 500(1.12)^8 \approx 500 \times 2.476 \approx 1{,}238 \text{ bacteria}$$

Example 2: Radioactive decay

A radioactive substance has a half-life of 25 years. Starting with 400 grams, how much remains after 100 years?

The model:

$$P(t) = 400 \left(\frac{1}{2}\right)^{t/25}$$

After 100 years, that’s 100/25 = 4 half-lives:

$$P(100) = 400 \left(\frac{1}{2}\right)^4 = 400 \times \frac{1}{16} = 25 \text{ grams}$$

Example 3: Solving for time (investment)

How long will it take $10,000 to grow to $25,000 at 6% compounded annually?

Set up the equation:

$$25{,}000 = 10{,}000(1.06)^t$$

Divide both sides by 10,000:

$$2.5 = (1.06)^t$$

Take the log of both sides:

$$t = \frac{\log 2.5}{\log 1.06} \approx \frac{0.3979}{0.0253} \approx 15.73 \text{ years}$$

Real-World Application

Exponential and logarithmic models are used in:

• Finance (compound interest, loan amortization, rule of 72)
• Biology and medicine (population growth, drug concentration decay)
• Physics (radioactive decay, Newton’s law of cooling)
• Environmental science (resource depletion, pollution spread)
• Epidemiology (disease spread modeling)

Example: Carbon-14 dating uses exponential decay to determine the age of ancient artifacts. The half-life of C-14 is about 5,730 years, so measuring how much remains tells you how old the sample is.

You’ve Got This

When modeling with exponentials, first identify whether it’s growth (base greater than 1) or decay (base between 0 and 1). For solving for time, take logarithms of both sides. These models are extremely useful in the real world. Practice turning word problems into equations, and you’ll see how powerful they are.

Quiz

$P(t) = 200(1.05)^t$ is an example of:

• A.Exponential decay
• B.Linear growth
• C.Exponential growth
• D.Quadratic growth

A substance has a half-life of 10 years. After 30 years, the remaining amount is:

• A.$\frac{1}{4}$ of the original
• B.$\frac{1}{8}$ of the original
• C.$\frac{1}{2}$ of the original
• D.$\frac{3}{4}$ of the original

To solve for $t$ in $P = P_0(1 + r)^t$, you should:

• A.Square both sides
• B.Divide both sides by $t$
• C.Add $t$ to both sides
• D.Take the logarithm of both sides

Which situation is best modeled by exponential decay?

• A.Radioactive half-life
• B.Linear salary increase
• C.Quadratic profit
• D.Constant speed

Compound interest is an example of:

• A.Exponential decay
• B.Linear growth
• C.A piecewise function
• D.Exponential growth