Exponential growth

Exponential and logarithmic growth: Exponential growth

Exponential growth

If after each time interval of fixed length, a value \(c\) changes by the same percentage, then \(c\) is actually multiplied by the same factor each time. We are then dealing with a constant multiple of an exponential function.

Exponential growth function

An exponential growth function has function rule \[f(x) = c \cdot g^x\]

Here,

\(c\) is the initial value, that is, the function value at \(x=0\),
\(g\) is the growth factor or the base,
\(x\) is the argument of the exponent

A growth that can be described by an exponential growth function, is also called an exponential growth model or simply exponential growth.

The exponential growth function is defined only for \(g \gt 0\).

This affects the range of the exponential growth function:

If \(c \gt 0\), then \(f(x) \gt 0\) for all \(x\)
If \(c \lt 0\), then \(f(x) \lt 0\) for all \(x\)

Here, \(y=0\) is a horizontal asymptote if \(g \neq 1\). This means the function never assumes the value #0#, while the function values come arbitrarily close to it. The function value comes closer and closer to #0# as #x# grows farther and farther from #0#. In the case #g \gt 1#, the function value approaches #0# when x goes away from #0# in the negative direction. In the case #0<g<1#, the function value approaches #0# when #x# grows farther and farther away from #0# in the positive direction.

If \(c = 0\), then #f(x)# is the constant function #0#; that is, \(f(x) = 0\) for all \(x\)

The argument #x# is the duration of growth. If #x# is a natural number, then #x# is the number of times \(c\) must be multiplied by the growth factor \(g\) in order to obtain the function value.

The exponential growth function #f(x) = c \cdot g^x# is an exponential function if and only if the constant #c# is equal to #1#.

In order to discuss the characteristics of an exponential growth function, we introduce two concepts that are applicable to arbitrary functions.

Growth and relative growth

Let #a# and #p# be numbers with #p\ge0# and let #f(x)# be a function defined for all #x# with #a\le x\le a+ p#.

The value #f(a+p)-f(a)# is called the growth of #f(x)# between #a# and #a+p#
If #f(a)\ne0#, then the value #\frac{f(a+p)-f(a)}{f(a)}# is called the relative growth of #f(x)# between #a# and #a+p#.

For example, if #f(x)=x^2#, then the growth of #f(x)# between #0# and #1# is equal to #f(1)-f(0)=1#. The relative growth of this function between #0# and #1# is not defined because #f(0)=0#. The relative growth of #f(x)# between #1# and #1+p# is equal to \(\frac{(1+p)^2-1}{1}=p\cdot(p+2)\).

The relative growth should not be confused with the difference quotient #\frac{f(a+p)-f(a)}{p}# that plays a major role in differentiation. In differentiation, the denominator is the length of the interval #\ivcc{a}{a+p}# while in the case of relative growth the denominator is the value of #f# at the starting point of the interval.

Below are some characteristics of exponential growth.

Relative growth of the exponential growth model

Let #g# be a positive number and #c# a constant distinct from #0#.

If \(f(x) = c\cdot g^x\), then the relative growth of #f(x)# between each pair of points #a# and #a+p# is equal to \[g^p-1\]
The relative growth rate of an exponential growth function is independent of the value of \(a\).
The relative growth rate of an exponential growth function is independent of the initial value \(c\).

1. The relative growth of #f(x)# between #a# and #a+p# is equal to \[\frac{f(a+p)-f(a)}{f(a)} =\frac{c\cdot g^{a+p}-c\cdot g^a}{c\cdot g^a}= g^p-1\]

2 and 3. The function rule of the relative growth rate is #g^p-1#. The variables #a# and #c# do not appear in this expression. This function rule only depends on the growth factor #g# and the length #p# of the interval #\ivcc{a}{a+p}#.

The first statement is a characteristic of exponential growth: If \(f(x)\) is a function of relative growth #g^p-1# between #a# and #a+p# for every pair of numbers #a# and #p# with #p\ge0#, then #f(x)# is of the form

\[f(x)=f(0)\cdot g^x\]

This can be seen by taking #a=0# and #p=x# in the formula of the definition of relative growth:

\[\frac{f(x)}{f(0)}-1 =\frac{f(x)-f(0)}{f(0)}=\frac{f(a+p)-f(a)}{f(a)}=g^p-1\]

This gives \(\frac{f(x)}{f(0)}=g^p\), so \(f(x)=f(0)\cdot g^x\).

This shows that the exponential growth model includes all the functions that, at each interval #\ivcc{a}{a+p}#, have relative growth #g^p-1#.

If \(y = c\cdot g^x\), then, as follows from the definition, the relative growth of the exponential growth model between \(a\) and \(a+1\) is equal to \(g-1\). We give this special value its own name:

Growth rate

The growth rate of an exponential growth with function rule \(f(x) = c \cdot g^x\) is equal to \(g-1\).

If we multiply the growth rate by \(100\), we get the fixed percentage increase or decrease of the exponential growth model.

Thus, the growth rate is the relative growth of the exponential growth model on an interval of the length #1#.

As we have seen in the definition of exponential growth function, it depends on the size of #g# whether there is an increase or decrease.

Describe all the discussed concepts for exponential growth with initial value \(5\) and growth factor \(4\).

For an exponential function with initial value \(5\) and growth factor \(4\) we have \(y = 5\cdot 4^{x}\).

If \(x\) increases from \(2\) to \(3\), then \(y\) increases from \(80\) to \(320\). Thus, the growth of \(y\) on the interval #\ivcc{2}{3}# is \({320-80} = 240\).

Consequently, the relative growth of \(y\) for this increase of \(x\) is equal to \( \frac{320-80}{80} =\frac{240}{80} = 3\).

But also, for every other increase of \(x\) by \(1\), the relative growth of \(y\) equals \( {3} \). After all, the growth rate of this growth model is equal to \( {4}-1 = {3} \).

The fixed percentage increase of this growth model is \(3 \cdot 100 = 300\)%.

The graph of the exponential growth function #5\cdot 4^{x}# is shown below. The large increase illustrates that exponential growth represents very strong growth.

New example