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All courses Algebra I · F-IF.8.b 70 of 76
Use exponent properties to interpret exponential expressions, including percent growth and decay.

Interpret the base of an exponential growth model as a percent increase

Problem
For the growth model \(P(t)=500(1.06)^t\), what does the base tell you about the percent increase each interval?
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Hint

Look at the base \(1.06\) and rewrite it as \(1 + 0.06\).

In an exponential growth model \(a(b)^t\), the base \(b\) is the growth factor, and \(b-1\) gives the decimal increase each interval.

Solution walkthrough

01

Identify the exponential base

\[\begin{aligned} P(t)=500(1.06)^t \\ \text{growth}~\text{factor}~b=1.06 \end{aligned}\]

In an exponential model a(b)t, the base b is the multiplier applied during each equal time interval.

02

Separate one whole from the change

\[1.06=1+0.06\]

The 1 represents retaining 100% of the current amount, while the additional 0.06 represents growth.

03

Convert the decimal change to a percent

\[0.06*100\%=6\%\]

Multiplying a decimal rate by 100 converts it to percent units.

04

Interpret in context

\[\text{multiply}~by~1.06~\text{each}~\text{interval}~->~6\%~\text{increase}~\text{each}~\text{interval}\]

Because the factor exceeds 1, the change is an increase. The initial value 500 does not alter the percent rate.

+

Another way

  1. Use rate=b-1=1.06-1=0.06=6%.

!

Common mistake

Do not call the change 106%. The factor is 106% of the previous amount, which means the amount increases by only the 6% above 100%.

Solution walkthrough video