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

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

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

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

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

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

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

5.1 Exponential Functions Homework pg 482 1-68

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

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

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

5.2 Logarithms and Logarithmic Functions Homework pg 491 1-94

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

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”

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

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

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,

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

5.3 The Exponential Function and Natural Logarithms Homework pg 502 1-106

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

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

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

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

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

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

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

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)= 1500 + 315 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.

5.4 Exponential/Logarithmic Equations and Applications Homework pg 516 1-106

Chapter 5 Review