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Section 5.3 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 5.25.

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 5.26.

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 \((1,0)\text{.}\)