Exponential Functions: Definitions, Theorems, and Solving Equations and Inequalities, Slides of Mathematics

Download Exponential Functions: Definitions, Theorems, and Solving Equations and Inequalities and more Slides Mathematics in PDF only on Docsity!

Exponential Functions

Listed below are exponential expressions. Which are: *exponential functions? *exponential equations? *exponential inequalities?

Definitions and Theorems. Definitions Let a ≠ 0. We define the following: (1) a

= 1 (2) a

-n

=

Theorems  Let r and s be rational numbers. Then (1)a r a s = a r+s (4) (ab) r = a r b r (2) = a r-s = (3)(a r ) s = a rs

Example 1. Solve the equation = 16. Solution: We write both sides with 4 as the base. Alternate Solution: We can also write both sides with 2 as the base.

 Solution: Both 125 and 25 can be written using 5 as the base.

Example 2. Solve the equation =

Property of Exponential

Inequalities

If b > 1, then the exponential function

y = b

x

is increasing for all x. This

means that b

x

< b

y

if and only if x < y.

If 0 < b < 1, then the exponential

function y = b

x

is decreasing for all x.

x y SOLVING EXPONENTIAL INEQUALITIES

Property of Exponential Inequalities This simply means that in solving exponential inequalities such as b m < b n , the resulting direction of the inequality (m < n or m > n) is based on whether the base b greater than 1 or less than

 (^) Solution: Both 9 and 3 can be written using 3 as the base. Example 1. Solve the inequality <. Since the base 3 > 1, then the direction of the inequality will be retained. Thus, the solution set is (4, +∞). You can verify that x = 5 and 6 are solutions, but x = 4 and 3 are not.

 S

Example 2. Solve the inequality ≥. Since the base < 1, then the direction of the inequality will be reversed. Thus, the solution set is [1, +∞). You can verify that x = 1 and 2 are solutions, but x = 0 and - are not.

Well done! You may now answer the Assignment.