Chapter 3: Exponential, Logistic, and Logarithmic Functions

Overview of Chapter 3 So far in this course, we have mostly studied algebraic functions, such as polys, rationals, and power functions w/ rat’l exponents… The three types of functions in this chapter (exponential, logistic, and logarithmic) are called transcendental functions, because they “go beyond” the basic algebra operations involved in the aforementioned functions…

Consider these problems: Evaluate the expression without using a calculator

We begin with an introduction to exponential functions: First, consider: This is a familiar monomial, and a power function… one of the “twelve basics?” Now, what happens when we switch the base and the exponent ??? This is an example of an exponential function

Definition: Exponential Functions Let a and b be real number constants. An exponential function in x is a function that can be written in the form where a is nonzero, b is positive, and b = 1. The constant a is the initial value of f (the value at x = 0), and b is the base. Note: Exponential functions are defined and continuous for all real numbers!!!

Identifying Exponential Functions Which of the following are exponential functions? For those that are exponential functions, state the initial value and the base. For those that are not, explain why not. 1. Initial Value = 1, Base = 3 2. Nope!  g is a power func.! 3. Initial Value = –2, Base = Initial Value = 7, Base = 1/2 5. Nope!  q is a const. func.!

More Practice with Exponents Given 1., find an exact value for:

Determine the formula for the exp. func. g: Finding an Exponential Function from its Table of Values xg(x)g(x) –24/9 –14/ The Pattern? x 3 General Form: Initial Value: Solve for b: Final Answer:

Determine the formula for the exp. func. h: Finding an Exponential Function from its Table of Values xh(x)h(x) –2128 – /2 The Pattern? x 1/4 General Form: Initial Value: Solve for b: Final Answer:

How an Exponential Function Changes (a recursive formula) For any exponential function and any real number x, If a > 0 and b > 1, the function f is increasing and is an exponential growth function. The base b is its growth factor. If a > 0 and b < 1, f is decreasing and is an exponential decay function. The base b is its decay factor. Does this formula make sense with our previous examples?