We will examine situations where quantities are increasing exponentially. This situation is known as $\textit{exponential growth modelling}$, and frequently occurs in real life around us.

The population of species, people, bacteria and investment usually $\textit{growth}$ in an exponential way.

Growth is exponential when the quantity present is multiplied by a constant for each unit time interval. this constant is called the $\textit{growth}$ or $\textit{compounding factor}$ is greater than one.

Considering a situation where the changes to a certain population of bacteria over time is a good example of $\textit{exponential growth}$.

It can be seen that the population is increasing but also that the growth rate is increasing; that is, the graph is getting steeper.

Consider a population of $10$ bacteria which, under favourable conditions, increase by $20\%$ each day.

To increase a quantity by $20\%$, it is known to multiply it by $1.2$.

If $B_n$ is the population of bacteria after $n$ days, then:

\( \begin{align} \displaystyle

B_0 &= 10 &\textit{the original population} \\

B_1 &= B_0 \times 1.2 = 10 \times 1.2 \\

B_2 &= B_1 \times 1.2 = 10 \times 1.2 \times 1.2 = 10 \times 1.2^2 \\

B_3 &= B_2 \times 1.2 = 10 \times 1.2^2 \times 1.2 = 10 \times 1.2^3 \\

\end{align} \)

and so on.

The population is $1.2$ times every day, so the $\textit{growth}$ or $\textit{compounding factor}$ is $2$.

The pattern above shows that $B_n = 10 \times 1.2^n$.

In general, the exponential growth function has an equation of the form: $y=Aa^{kx}$

- $A$, $a$ and $k$ are constants.
- $a>1$ and $k>0$ are the growth or compounding factor.
- $A$ is the initial value of $y$ (when $x=0$)

### Example 1

The weight $B_n$ of bacteria in a colony $n$ hours after establishment is given by $B_n = 100 \times 5^{0.2n}$ grams.

(a) Find the initial weight.

\( \begin{align} \displaystyle

B_0 &= 100 \times 5^{0.2 \times 0} \\

&= 100 \times 5^0 \\

&= 100 \times 1 \\

&= 100

\end{align} \)

(b) Find the weight after $5$ hours.

\( \begin{align} \displaystyle

B_5 &= 100 \times 5^{0.2 \times 5} \\

&= 100 \times 5^1 \\

&= 100 \times 5 \\

&= 500

\end{align} \)

(c) Find the percentage increase from $n=10$ to $n=20$.

\( \begin{align} \displaystyle

\dfrac{B_{20}-B_{10}}{B_{10}} \times 100\% &= \dfrac{100 \times 5^{0.2 \times 20}-100 \times 5^{0.2 \times 10}}{100 \times 5^{0.2 \times 10}} \times 100 \% \\

&= \dfrac{5^{0.2 \times 20}-5^{0.2 \times 10}}{5^{0.2 \times 10}} \times 100 \% \\

&= \dfrac{5^4-5^2}{5^2} \times 100 \% \\

&= \dfrac{5^2(5^2-1)}{5^2} \times 100 \% \\

&= (5^2-1) \times 100 \% \\

&= 2400 \%

\end{align} \)

### Example 2

The speed $V_t$ is given by $V_t = 8 \times 2^{0.5t}$, where $t$ is the temperature in $^{\circ}C$. Find the temperature when the speed is $24$, correcting to three significant figures.

\( \begin{align} \displaystyle

8 \times 2^{0.5t} &= 24 \\

2^{0.5t} &= 3 \\

0.5t &= \log_2{3} \\

t &= \dfrac{1}{0.5} \log_2{3} \\

&= 3.1699 \cdots \\

&= 3.17 ^{\circ}C

\end{align} \)

Algebra Algebraic Fractions Arc Binomial Expansion Capacity Common Difference Common Ratio Differentiation Double-Angle Formula Equation Exponent Exponential Function Factorials Factorise Functions Geometric Sequence Geometric Series Index Laws Inequality Integration Kinematics Length Conversion Logarithm Logarithmic Functions Mass Conversion Mathematical Induction Measurement Perfect Square Perimeter Prime Factorisation Probability Product Rule Proof Quadratic Quadratic Factorise Ratio Rational Functions Sequence Sketching Graphs Surds Time Transformation Trigonometric Functions Trigonometric Properties Volume