Math Exponential functions Use of exponential functions

Use of exponential functions

Exponential functions are used to describe exponential growth and decay processes.

Exponential growth and decay have the following function equation:

$N(t)=a\cdot b^t$

$t...$ time
$a ...$ initial level
$b ...$ growth factor
$N(t) ...$ value in dependence on $t$

Exponential growth and decay



  • If the growth factor $b$ is greater than 1, it is an exponential growth (increase).
  • If the growth factor $b$ is less than 1 and greater than 0, it is an exponential decay (decrease).


One person takes 6mg of a drug. It is broken down by $\frac13$ each day. How much mg is still in the body after 7 days?

  1. Determine values from the task

    Every day $\frac13$ is broken down, which means $\frac23$ is left. The decrease factor is $b=\frac23$
    The initial level $a=6$
    The 7th day is of interest. Therefore, the time is $t=7$
  2. Insert values in the formula and calculate

    $N(\color{purple}{t})=\color{red}{a}\cdot \color{green}{b}^\color{purple}{t}$
    $N(\color{purple}{7})=\color{red}{6}\cdot (\color{green}{\frac23})^\color{purple}{7}\approx0.35$
    On the 7th day there is still about 0.35 mg of the drug in the body.

Percentage growth

If there is an increase or decrease with a constant percentage growth factor, it is percentage growth. This is also an exponential growth process and can therefore be described with an exponential function.

When increasing with a percentage growth rate $p$ % growth factor $(1+\frac{p}{100})$ applies and when decreasing with a percentage decrease rate $p$ % the decrease factor $(1-\frac{p}{100})$ applies.

Substituting the growth or decrease factor into the above formula for $b$, one obtains with a percentage increase the equation

$N(t)=a\cdot (1+\frac{p}{100})^t$

and at percentage decrease the equation

$N(t)=a\cdot (1-\frac{p}{100})^t$


3m² of the water surface of a lake are covered with a type of algae, which grows annually by 50%. Calculate the covered area for the next 5 years.

  1. Determine values from the task

    The percentage growth rate is $p=50$
    The initial level $a=3$
    For the time $t$ , the years 1 to 5 must be used.
  2. Insert values in the formula

    This is a percentage increase as there is a percentage growth factor and the area is growing. First, we put the constant values $p$ and $a$ into the percentage increase formula.

    $N(\color{purple}{t})=\color{red}{a}\cdot (1+\frac{\color{green}{p}}{100})^\color{purple}{t}$
    $N(\color{purple}{t})=\color{red}{3}\cdot (1+\frac{\color{green}{50}}{100})^\color{purple}{t}=\color{red}{3}\cdot1.5^\color{purple}{t}$

    Now use the values 1 to 5 for $t$ and calculate.