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All courses Math II · F-IF.8.b 24 of 73
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?
Your answer
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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

\[P(t)~=~500(1.06)^t\]

The base of the exponential part is \(1.06\).

02

Interpret the growth factor

\[\text{growth}~\text{factor}~=~1.06\]

This means the quantity is multiplied by \(1.06\) each interval.

03

Convert the base to percent increase

\[1.06~=~1+0.06\]

The extra \(0.06\) represents a \(6\%\) increase each interval.

04

State what the base tells you

\[\text{growth}~\text{factor}:~1.06;~\text{percent}~\text{change}:~6\%~\text{increase}~\text{per}~\text{interval}\]

So the base shows the quantity grows by \(6\%\) every interval.

+

Another way

  1. Subtract \(1\) from the base and convert to a percent: \(1.06-1=0.06\), so the increase is \(6\%\).

!

Common mistake

Reading the base \(1.06\) as \(106\%\) increase instead of recognizing that the increase is only the part above \(1\).