# 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 $\textit{logarithms}$.

$500$ is invested in an account that pays $4.5\%$ per annum, interest compounded monthly. Find how long it takes to reach $\$5000. \begin{align} \displaystyle A_{n+1} &= 5000 \\ A_{1} &= 5000 \\ r &= 104.5\% \\ &= 1.045 \\ A_{n+1} &= A_{1} \times r^{n} \\ 5000 &= 500 \times 1.045^{n} \\ 1.045^{n} &= \dfrac{5000}{500} \\ &= 10 \\ n &= \log_{1.045}{10} \\ &= \dfrac{\log_{10}{10}}{\log_{10}{1.045}} \\ &= 52.311 \cdots \\ \end{align} Therefore it takes53$days. ## Decay ### Example 3 The mass$M_{t}$of radioactive substance remaining after$y$years given by$M_{t}=6000 \times e^{-0.05t}grams. Find the time taken for the mass to halve. \begin{align} \displaystyle 6000 \times e^{-0.05t} &= 3000 \\ e^{-0.05t} &= 3000 \div 6000 \\ &= 0.5 \\ -0.05t &= \log_{e}{0.5} \\ t &= -\dfrac{1}{0.05}\log_{e}{0.5} \\ &= 13.862 \cdots \end{align} Therefore it takes around14$years. Unlock your full learning potential—download our expertly crafted slide files for free and transform your self-study sessions! Discover more enlightening videos by visiting our YouTube channel! ### Related Articles ## High School Math for Life: Making Sense of Earnings Salary Salary refers to the fixed amount of money that an employer pays an employee at regular intervals, typically on a monthly or biweekly basis,… ## Mastering Integration by Parts: The Ultimate Guide Welcome to the ultimate guide on mastering integration by parts. If you’re a student of calculus, you’ve likely encountered integration problems that seem insurmountable. That’s… ## 12 Patterns of Logarithmic Equations Solving logarithmic equations is done using properties of logarithmic functions, such as multiplying logs and changing the base and reciprocals of logarithms.$\large \begin{aligned}… ## The Best Practices for Using Two-Way Tables in Probability Welcome to a comprehensive guide on mastering probability through the lens of two-way tables. If you’ve ever found probability challenging, fear not. We’ll break it… ## Binomial Expansions The suma+b$is called a binomial as it contains two terms.Any expression of the form$(a+b)^n\$ is called a power of a binomial. All…

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