Skip to main content

Section 10.3 The Natural Base

Subsection 1. Using growth and decay laws with base \(e\)

Subsubsection Examples

We can write exponential growth and decay laws using bas \(e\text{.}\)

Exponential Growth and Decay.

The function

\begin{equation*} P(t) = P_0 e^{kt} \end{equation*}

describes exponential growth if \(k \gt 0\text{,}\) and exponential decay if \(k \lt 0\text{.}\)

Example 10.24.

A colony of bees grows at a rate of 8% annually. Write its growth law using base \(e\text{.}\)

Solution

The growth factor is \(~b = 1+r = 1.08~\text{,}\) so the growth law can be written as

\begin{equation*} P(t) = P_0 (1.08)^t \end{equation*}

Using base \(e\text{,}\) we write \(~P(t) = P_0 e^{kt},~\) where \(e^k = 1.08.\) (You can see this by evaluating each growth law at \(t=1\text{.}\)) So we solve for \(k\text{.}\)

\begin{align*} e^k \amp = 1.08 \amp \amp \blert{\text{Take the natural log of both sides.}}\\ \ln (e^k) \amp = \ln (1.08) \amp \amp \blert{\text{Simplify both sides.}}\\ k \amp = 0.0770 \end{align*}

The growth law is \(~P(t) = P_0 e^{0.077t}\text{.}\)

Example 10.25.

A radioactive isotope decays according to the formula \(~N(t)=N_0 e^{-0.016t},~\) where \(t\) is in hours. Find its percent rate of decay.

Solution

First we write the decay law in the form \(~N(t)=N_0 b^t,~\) where \(~b=e^k.~\)

In this case, \(~k=-0.016,~\) so \(~b=e^{-0.016} = 0.9841.~\) Now, \(~b=1-r,~\) and solving for \(r\) we find \(~r=-0.0159.~\) The rate of decay is approximately 16% per hour.

Subsubsection Exercises

A virus spreads in the population at a rate of 19.5% daily. Write its growth law using base \(e\text{.}\)

Answer
\(P(t) = P_0 e^{0.178t}\)

Sea ice is decreasing at a rate of 12.85% per decade. Write its decay law using base \(e\text{.}\)

Answer
\(Q(t) = Q_0 e^{-0.1375t}\)

In 2020, the world population was growing according to the formula \(~P(t)=P_0 e^{0.0488t},~\) where \(t\) is in years. Find its percent rate of growth.

Answer
5%

Since 1984, the population of cod has decreased annually according to the formula \(~N(t)=N_0 e^{-0.1863t}.~\) Find its percent rate of decay.

Answer
17%

Subsection 2. Graphing \(y=e^x\) and \(y=\ln x\)

The graphs of the natural exponential function and the natural log function have some special properties.

Subsubsection Exercises

Use technology to graph \(~f(x)=e^x~\) and \(~y=x+1~\) in a window with \(~-2 \le x \le 3~\) and \(~-1 \le y \le 4~\text{.}\) What do you notice about the two graphs?

Answer
graph

The line is tangent to the graph at \((0,1)\text{.}\)

Use technology to graph \(~f(x)=\ln x~\) and \(~y=x-1~\) in a window with \(~-1 \le x \le 4~\) and \(~-2 \le y \le 3~\text{.}\) What do you notice about the two graphs?

Answer
graph

The line is tangent to the graph at \((1,0)\text{.}\)