Education Technology

Calculus: Exponential Functions and the Natural Logarithm

by Texas Instruments


  • Understand the distinction between growth rate (slope) and relative growth rate (slope/y -value) of a function
  • Understand that the relative growth rate f'(x) / f(x) is constant for exponential functions of the form f(x) = bx
  • Iidentify the natural logarithm of the base of an exponential function as this constant relative growth rate
  • Describe the behavior of the graph of y = f(x) = bx for different values of b
  • Understand what is special about the number e as a base for exponential functions in terms of growth rate


  • Relative growth rate
  • Exponential function
  • Natural logarithm
  • Natural exponential function and its base e

About the Lesson

The purpose of this lesson is to guide students to discover a surprising property involving the relative growth rate of an exponential function. Students will consider the ratio f'(x) / f(x) for f(x) = bx and see that for a given value b this ratio is a constant for all values of x. We define this ratio as the natural logarithm of b, denoted ln(b).