1. Exponential and Logarithmic Functions
Chapter 5
Exponential and Logarithmic Functions
We’ve talked about several different types of functions and now we get to talk about two new types

2. For b > 0, b≠1, f(x) = bx defines the base b exponential function.
5.1 Exponential Functions
Exponential Functions
For b > 0, b≠1, f(x) = bx defines the base b exponential function.
The domain of f is all real numbers.
b has to be greater than 0 otherwise we could end up taking square roots of negative numbers. Ex. F(x) = (-4)^1/2
b can also not equal zero due to the fact that it would generate a constant function because 1 to any power is still equal to 1

3. Exponential Properties
5.1 Exponential Functions
Exponential Properties
Given a, b, x, and t are real numbers, with b, c > 0,
All of our regular properties will still hold true for rational and irrational exponents

4. Graphs of exponential functions
Important Characteristics
One-to-one function
Domain:
Y-intercept (0,1)
Range:

5. Decreasing if 0<b<1
5.1 Exponential Functions
Increasing if b>1
Decreasing if 0<b<1
b^x and b^-x are reflections across the y-axis

6. EXPONENTIAL EQUATIONS WITH LIKE BASES THE UNIQUENESS PROPERTY
5.1 Exponential Functions
EXPONENTIAL EQUATIONS WITH LIKE BASES
THE UNIQUENESS PROPERTY
If bm = bn, then m = n.
If m = n, then bm = bn.
Important properties to help solve exponential equaitons

7. EXPONENTIAL EQUATIONS WITH LIKE BASES THE UNIQUENESS PROPERTY
5.1 Exponential Functions
EXPONENTIAL EQUATIONS WITH LIKE BASES
THE UNIQUENESS PROPERTY
First step is to rewrite each side using the same base for the exponents. (sometimes it may be necessary to rewrite only one side while other times you may have to rewrite both sides.
Second step is to use properties of exponents to write each side with the single base and single exponent.
Third step is to use the uniqueness property to write just the exponents as equal to each other. If the bases are the same then what they are being raised to must be the same also.
Fourth step is to solve the resulting equation for the variable

8. 5.1 Exponential Functions
Homework pg

9. Logarithmic Functions
5.2 Logarithms and Logarithmic Functions
Logarithmic Functions
For b > 0, b ≠ 1, the base-b logarithmic function is defined as
Write in exponential form
Write in logarithmic form

10. Graphing Logarithmic Functions Calculators and Common Logarithms
5.2 Logarithms and Logarithmic Functions
Graphing Logarithmic Functions
Calculators and Common Logarithms

11. Pg 493 #87 and 88 Earthquake Intensity
5.2 Logarithms and Logarithmic Functions
Pg 493 #87 and 88
Earthquake Intensity

12. 5.2 Logarithms and Logarithmic Functions
Homework pg

13. Natural Logarithmic Function
5.3 The Exponential Function and Natural Logarithms
Natural Logarithmic Function

14. Properties of Logarithms
5.3 The Exponential Function and Natural Logarithms
Properties of Logarithms
Given M, N, and b are positive real numbers, where b ≠ 1, and any real number x.
Product Property:
“the log of a product is equal to a sum of logarithms”
Quotient Property:
“The log of a quotient is equal to a difference of logarithms”
Power Property:
“The log of a number to a power is equal to the power times the log of the number”

15. Using Properties of Logarithms
5.3 The Exponential Function and Natural Logarithms
Using Properties of Logarithms

16. Using Properties of Logarithms
5.3 The Exponential Function and Natural Logarithms
Using Properties of Logarithms

17. Given the positive real numbers M, b, and d, where b≠1 and d≠1,
5.3 The Exponential Function and Natural Logarithms
Change of Base Formula
Given the positive real numbers M, b, and d, where b≠1 and d≠1,

18. Using the change of base formula
5.3 The Exponential Function and Natural Logarithms
Using the change of base formula

19. 5.3 The Exponential Function and Natural Logarithms
Homework pg

20. Writing Logarithmic and Exponential Equations in Simplified Form
5.4 Exponential/Logarithmic Equations and Applications
Writing Logarithmic and Exponential Equations in Simplified Form

21. Writing Logarithmic and Exponential Equations in Simplified Form
5.4 Exponential/Logarithmic Equations and Applications
Writing Logarithmic and Exponential Equations in Simplified Form

22. Solving Exponential Equations
5.4 Exponential/Logarithmic Equations and Applications
Solving Exponential Equations
For any real numbers b, x, and k, where b>0 and b≠1

23. Solving Exponential Equations
5.4 Exponential/Logarithmic Equations and Applications
Solving Exponential Equations

24. Solving Exponential Equations
5.4 Exponential/Logarithmic Equations and Applications
Solving Exponential Equations

25. Solving Logarithmic Equations
5.4 Exponential/Logarithmic Equations and Applications
Solving Logarithmic Equations
For real numbers b, m, and n where b > 0 and b≠1,
Equal bases imply equal arguments

26. Solving Logarithmic Equations
5.4 Exponential/Logarithmic Equations and Applications
Solving Logarithmic Equations
Use quadratic formula to solve for x

27. 5.4 Exponential/Logarithmic Equations and Applications
An advertising agency determines the number of items sold is related to the amount spent on advertising by the equation N(A)= ln A, where A represents the advertising budget and N(A) gives the number of sales. If a company wants to generate 5000 sales, how much money should be set aside for advertising? Round interest to the nearest dollar.

28. 5.4 Exponential/Logarithmic Equations and Applications
Homework pg

29. Chapter 5 Review