Skill: Write exponential equations from contexts

A situation has exponential change when a quantity is multiplied by the same factor during each time period. Use the model

$$y=ab^t$$

where:

• $a$ is the initial value, when $t=0$;
• $b$ is the multiplicative change factor;
• $t$ is the number of elapsed time periods;
• $y$ is the amount after $t$ periods.

For a percent increase of $r$,

$$b=1+r.$$

For a percent decrease of $r$,

$$b=1-r.$$

Remember to write the percent as a decimal before using it.

Example

A town has a population of $12{,}000$. Its population increases by $3\%$ each year. Write an exponential equation for the population $P$ after $t$ years.

Step 1: Identify the initial value.

The population starts at $12{,}000$, so

$$a=12{,}000.$$

Step 2: Find the growth factor.

A $3\%$ increase means

$$r=0.03.$$

Therefore,

$$b=1+0.03=1.03.$$

This means the population is multiplied by $1.03$ each year.

Step 3: Identify the time variable.

Since the change happens each year, let $t$ represent the number of years.

Step 4: Substitute into the exponential model.

$$\boxed{P=12{,}000(1.03)^t}$$

Here, $P$ is the population and $t$ is measured in years.

For example, after $5$ years,

$$P=12{,}000(1.03)^5.$$

The expression is exponential because the population is multiplied by the same factor, $1.03$, each year. It is not modeled by $12{,}000+0.03t$, because that would represent a constant additive change rather than a constant percent change.