#### Objectives

- 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*) =*b*^{x} - 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*) =*b*^{x}for different values of*b* - Understand what is special about the number
*e*as a base for exponential functions in terms of growth rate

#### Vocabulary

- 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*) = *b*^{x} 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*).