2. Chapter 5: Exponential and Logarithmic Functions
5.1 Inverse Functions
5.2 Exponential Functions
5.3 Logarithms and Their Properties
5.4 Logarithmic Functions
5.5 Exponential and Logarithmic Equations and Inequalities
5.6 Further Applications and Modeling with Exponential and Logarithmic Functions

3. 5.2 Exponential Functions
Additional Properties of Exponents
For any real number a > 0, a  0, the following statements
are true:
(a) ax is a unique real number for each real
number x.
(b) ab = ac if and only if b = c.
(c) If a > 1 and m < n, then am < an.
If 0 < a < 1 and m < n, then am > an.

4. 5.2 Exponential Functions
If a > 0, a  1, then
f (x) = ax
defines the exponential function with base a.

5. 5.2 Graphs of Exponential Functions
Example Graph Determine the domain
and range of f.
Solution
There is no x-intercept. Any number to the zero
power is 1, so the y-intercept is (0,1). The domain
is (– ,), and the range is (0,).

6. 5.2 Comparing Graphs Example Explain why the graph of
is a reflection across the y-axis of the graph of
Analytic Solution
Show that g(x) = f (–x).

7. 5.2 Comparing Graphs Graphical Solution
The graph below indicates that g(x) is a reflection
across the y-axis of f (x).

8. 5.2 Graph of f (x) = ax, a > 1

9. 5.2 Graph of f (x) = ax, 0 < a < 1

10. 5.2 Using Translations to Graph an Exponential Function
Example Explain how the graph of
is obtained from the graph of
Solution

11. 5.2 Example using Graphs to Evaluate Exponential Expressions
Example Use a graph to evaluate
Solution With we find
that y  from the graph of y = 0.5x.

12. 5.2 Exponential Equations (Type I)
Example Solve
Solution
Write with the same base.
Set exponents equal and solve.

13. 5.2 Using a Graph to Solve Exponential Inequalities
Example Solve the inequality
Solution Using the graph below, the graph lies
above the x-axis for values of x less than 0.5.
The solution set for y > 0 is (–, 0.5).

14. 5.2 Compound Interest Recall simple earned interest where
P is the principal (or initial investment),
r is the annual interest rate, and
t is the number of years.
If A is the final balance at the end of each year, then

15. 5.2 Compound Interest Formula
Example Suppose that $1000 is invested at an annual
rate of 4%, compounded quarterly. Find the total amount in
the account after 10 years if no withdrawals are made.
Solution
The final balance is $
Suppose that a principal of P dollars is invested at an annual interest rate r (in decimal form), compounded n times per year. Then, the amount A accumulated after t years is given by the formula

16. 5.2 The Natural Number e
Named after Swiss mathematician Leonhard Euler
e involves the expression
e is an irrational number
Since e is an important
base, calculators are
programmed to find
powers of e.

17. 5.2 Continuous Compounding Formula
Example Suppose $5000 is deposited in an account
paying 3% compounded continuously for 5 years. Find the
total amount on deposit at the end of 5 years.
Solution
The final balance is $
If an amount of P dollars is deposited at a rate of interest r (in decimal form) compounded continuously for t years, then the final amount in dollars is

18. 5.2 Modeling the Risk of Alzheimer’s Disease
Example The chances of dying of influenza or pneumonia increase exponentially after age 55 according to the function defined by
where r is the risk (in decimal form) at age 55 and x
is the number of years greater than 55. Compare the
risk at age 75 with the risk at age 55.
Solution
x = 75 – 55 = 20, so
Thus, the risk is almost fives times as great at age 75
as at age 55.