# Exponential Growth and Decay using Logarithms

It has been known how exponential functions can be used to model various growth and decay situations. These included the growth of populations and the decay of radioactive substances. This lesson considers more growth and decay problems, focusing on how logarithms can be used in their solution.

## Population Growth

### Example 1

The area $A_{t}$ affected by the increase by the insects are given by $A_{t}=200 \times 2^{0.5t}$ m2, where $t$ is the number of days after the initial observation. Find the days taken for the affected area to reach $1000$ m2.

\begin{align} \displaystyle 200 \times 2^{0.5t} &= 1000 \\ 2^{0.5t} &= \dfrac{1000}{200} \\ &= 5 \\ 0.5t &= \log_{2}{5} \\ t &= \dfrac{1}{0.5}\log_{2}{5} \\ t &= 2 \times \dfrac{\log_{10}{5}}{\log_{10}{2}} \\ t &= 4.64 \cdots \\ \end{align}
Therefore it takes $5$ days.

## Financial Growth

A certain amount of $A_{1}$ is invested at a fixed rate for each compounding period in a financial situation. In this case, the value of the investment after $n$ periods is given by $A_{n+1}=A_{1} \times r^{n}$ where $r$ is the multiple corresponding to the given rate of interest. To find $n$ algebraically, it is required to use $\textit{logarithms}$.

## Mastering Exponential Growth: Your Ultimate Guide

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