# Exponential Growth and Decay Tutorial

## Introduction

### Equations

There are many situations where the increase or decrease of some variable in a fixed time interval will be proportional to the magnitude of the variable at the beginning of that time interval.

For example, let's look at a population of wee beasties which increases by 10% per year. If there were 100 wee beasties now, there would be an increase of 10 wee beasties after a year. We would see an increase of 500 wee beasties in a year when there were 5000 at the beginning.

Likewise, we can look at a population which decreases by 50% (i.e. a decrease to 1/2, or by a factor of 2) every day. A population of 100 would be down to 50 a day later, and a population of 5000 would drop to 2500 after one day.

These are all examples of exponential growth and decay.

A single equation can be used to solve all problems involving this type of change:

\(N = N_o \cdot e^{k \cdot t}\)

where '\(N\)' is the number in the population after a time '\(t\)', '\(N_o\)' is the **initial number**, '\(k\)' is the **growth constant** (if *positive*) or the **decay constant** (if *negative*), and 'e' is the base of natural logarithms (approximately 2.71828).

We can re-write this equation in another convenient form. Dividing the equation by \(N_o\), and then taking the natural logarithms of both sides, we get

\(ln \Bigl(\frac {N}{N_o}\Bigr) = k \cdot t\)

Note that in this form, we do not need to know the absolute values of '\(N\)' or '\(N_o\)'; all we need to know here is the ratio of these two values.

These two problems are used to solve questions for both exponential growth and exponential decay.

### Exponential Growth

### Exponential Decay

\(k = -0.112 day ^{-1}\)

### Factor

Before we go on to discuss exponential decay, we should pause and discuss some of the terms that are frequently used in these problems. First let's talk about the term **factor**.

Suppose you invested \(\$100\), and after a time, your investment was worth \(\$300\). The final value (\(\$300\)) would be three (3) times the initial value. We would say that your investment had *increased by a factor of 3*.

On the other hand, if you made a poor investment, and the value decreased from \(\$100\) to \(\$25\), then the final value would be a quarter (1/4) of the initial value. We would say the investment had *decreased by a factor of 4*.

If a population of 500 *increased* by a factor of 1.5, then there would be a population of (500)(1.5) = 750. If the same population *decreased* from 500 by a factor of 1.5, then there would be 500/1.5 = 333 remaining.

When there is an **increase**, the ratio of the final number to the initial number is the factor. That is

\(\frac {N_\mathrm {final}}{N_\mathrm {initial}} = \mathrm {the \; factor}\)

Similarly, if there is a decrease, the ratio of the final number to the initial number is the reciprocal of the factor:

\(\frac {N_\mathrm {final}}{N_\mathrm {initial}} =\frac {1}{ \mathrm {the \; factor}}\)

### Percentage Changes

You should be sure you understand percentage changes. The general equation for percentage changes is

\(\frac {N_{\mathrm final} - N_{\mathrm {initial}}}{N_{\mathrm {initial}}} \cdot (100) = \% \; \mathrm {change}\)

Thus, a $100 investment that increased to \(\$300\) (increased by a factor of 3), had a percentage increase of:

\(\frac {\$300 - \$100}{\$100} \cdot (100) = 200\%\)

The $100 investment that decreased to \(\$25\) (decreased by a factor of 4), had a percentage decrease of:

\(\frac {\$25 - \$100}{\$100} \cdot (100) = 75\%\)