In everyday speech, the term “exponential growth” is used rather loosely to describe any situation involving rapid growth. Many life science systems have an exponential relationship with time. For example: the number of bacteria in a culture may double every hour, an example of exponential growth. In this section we will be learning more about exponential growth.

When a quantity is increased by multiplier that is greater than $1$ over fixed periods of time then we have exponential growth.
The simple formula that describes exponential growth is

$y = a^{x}$

Where $a$ is a constant that depends on the system being studied, and $x$ is the exponent also called the index.

Taking the logarithm of both sides of the general equation gives

$\log y$ = $x \log a$

So, if we plot $\log y$ against $x$, an exponential relationship will plot as a straight line with a gradient of $\log a$.

The more general formula for exponential growth can be written in the form

$y = ae^{bx}$

where $a$ and $b$ are constants that depends on the system.

Taking the natural logarithm of both sides of the this equation gives:

$\ln y = \ln(ae^{bx})$

using logarithm rules for multiplication, this is the same as

$\ln y = ln a + bx ln e$

however we know that $\ln e =1$,

so $ln y = bx + ln a$
If the rate of change of a quantity, P, is proportional to P, then

$\frac{dP}{dt}$ = $bP$.....(1)
The solution to equation (1) is

$P = P_{0}e^{bt}$

$P_{0}$ is the initial value of P (i.e., when $t = 0$, $P = P_{0}$),
$b$ is the constant of proportionality.
$P$ is the amount of the quantity present at time $t$.
Graphs of exponential functions of the form

$y = ae^{bx}$

where $a$ and $b$ are constants, and $x \geq 0$

Exponential Growth

Every exponential function has the same shape as this function. The graph of an exponential decay function will be the reflection of an exponential growth function across the y-axis. Translating, reflecting, stretching, and contracting an exponential function will only change the scale or position of the function. These transformations will not change the shape of the function.

Exponential growth function can be any of the two below:

Exponential Growth Graph
The following are the example of exponential growth.

Solved Examples

Question 1: Solve for $x$, giving answer corrected to two decimal places.

$e^{x} = 5.47$

Solution:
 
Given $e^{x} = 5.47$

$x = \ln 5.47$

$e^{x} = a$; $x = log_{e}a$

$x= 1.70$  

 

Question 2: Solve for $x$, giving answer corrected to two decimal places.

$2e^{3x} = 3.72$

Solution:
 
Given $2e^{3x} = 3.72$

divide both sides by $2$

$e^{3x}$ = $\frac{3.72}{2}$

$3x$ = $\ln($$\frac{3.72}{2}$$)$

 $x$ = $\ln$ $\frac{\frac{3.72}{2}}{3}$

$x = 0.21$
 

Question 3: The current population in certain country is 8 million. what will be the population in 13 years, if the population grows at annual rate of 6%?

Solution:
 
Lets measure population in millions and time in years.

Given that, 
Time (t) = 13 years
initial population $P_{0} = 8$ million  
 
We know that, $P = P_{0}e^{bt}$  

$P = 8 e^{0.06\times 13}$ 

$P$ = $17.451$ million

The population in 13 years is $P$ ≈  $17.451$ million