Exponential Functions: Apply It 1

  • Evaluate exponential functions.
  • Find the equation of an exponential function.
  • Use compound interest formulas.

Applications of Exponential Functions

Exponential functions are incredibly powerful tools in mathematics, and they have a wide range of applications in the real world. Whether you’re looking at population growth, radioactive decay, or even finance, exponential functions help us model situations where change happens at a constant multiplicative rate.

Escherichia coli (e Coli) bacteria
An electron micrograph of E.Coli bacteria. (credit: “Mattosaurus,” Wikimedia Commons)

Bacteria commonly reproduce through a process called binary fission during which one bacterial cell splits into two. When conditions are right, bacteria can reproduce very quickly. Unlike humans and other complex organisms, the time required to form a new generation of bacteria is often a matter of minutes or hours as opposed to days or years.[1]

For simplicity’s sake, suppose we begin with a culture of one bacterial cell that can divide every hour. The table below shows the number of bacterial cells at the end of each subsequent hour. We see that the single bacterial cell leads to over one thousand bacterial cells in just ten hours! If we were to extrapolate the table to twenty-four hours, we would have over [latex]16[/latex] million!

Hour [latex]0[/latex] [latex]1[/latex] [latex]2[/latex] [latex]3[/latex] [latex]4[/latex] [latex]5[/latex] [latex]6[/latex] [latex]7[/latex] [latex]8[/latex] [latex]9[/latex] [latex]10[/latex]
Bacteria [latex]1[/latex] [latex]2[/latex] [latex]4[/latex] [latex]8[/latex] [latex]16[/latex] [latex]32[/latex] [latex]64[/latex] [latex]128[/latex] [latex]256[/latex] [latex]512[/latex] [latex]1024[/latex]
A function that models exponential growth grows by a rate proportional to the amount present. For any real number [latex]x[/latex] and any positive real numbers [latex]a[/latex] and [latex]b[/latex] such that [latex]b\ne 1[/latex], an exponential growth function has the form

[latex]\text{ }f\left(x\right)=a{b}^{x}[/latex]

where

  • [latex]a[/latex] is the initial or starting value of the function.
  • [latex]b[/latex] is the growth factor or growth multiplier per unit [latex]x[/latex].
The population of India was about [latex]1.25[/latex] billion in the year 2013 with an annual growth rate of about [latex]1.2 \%[/latex]. This situation is represented by the growth function [latex]P\left(t\right)=1.25{\left(1.012\right)}^{t}[/latex] where [latex]t[/latex] is the number of years since 2013. To the nearest thousandth, what will the population of India be in 2031?

  1. Todar, PhD, Kenneth. Todar's Online Textbook of Bacteriology. http://textbookofbacteriology.net/growth_3.html. ↵