Exponential and Logarithmic Functions Exponential Functions & Their Graphs Logarithmic Functions & Their Graphs Properties of Logarithms Exponential and Logarithmic Equations Exponential and Logarithmic Models

Exponential Functions Definition of Exponential Functions The exponential function f with base a is denoted by where a > 0, a ≠ 1, and x is any real number. Example 1 Evaluating Exponential Expressions Use a calculator to evaluate each expression. a. b. Rounded to nearest ten thousandth a. b.

Graphs of Exponential Functions The graphs of all exponential functions have similar characteristics. Example 2 Graphs of y = ax Create a table x -3 -2 -1 1 2 2x 4x Graph of y = 2x Graph of y = 4x Combined Graphs

Graphs of Exponential Functions The graphs of all exponential functions have similar characteristics. Example 2 Graphs of y = a- x Create a table x -3 -2 -1 1 2 2-x 4-x Graph of y = 2-x Graph of y = 4-x Combined Graphs

Comparing Exponential Functions Basic Characteristics of Exponential Functions Function Y = ax, a >1 Y = a-x, a >1 Domain: (- ∞, ∞) Range: (0, ∞) y-Intercept: (0, 1) (0,1) Increasing/Decreasing Increasing Decreasing Horizontal Asymptote y = 0 Continuity: Continuous Graph of y = ax Graph of y = a-x

Shifting Exponential Functions Each of the following graphs represents a transformation of the graph of f(x) = 3x Because g(x) = 3x+1 = f(x+1) , the graph of g can be obtained by shifting the graph of f one unit to the left. Because g(x) = 3x - 2 = f(x) - 2 , the graph of g can be obtained by shifting the graph of f one unit to the left.

Shifting Exponential Functions Each of the following graphs represents a transformation of the graph of f(x) = 3x Because g(x) = 3x - 2 = f(x) - 2 , the graph of g can be obtained by shifting the graph of f one unit to the left. Because g(x) = - 3x = - f(x) , the graph of g can be obtained by shifting the graph of f one unit to the left.

The Natural Base e In many applications the most convenient choice for a base is the irrational number e. e ≈ 2.71828. . . This number is known as the natural base. The function f(x) = ex is called the natural exponential function. Example 3 Evaluating the Natural Exponential Function Graph of the Natural Base Use a calculator to evaluate each expression. a. b. c. d. Rounded to nearest ten thousandth a. b. c. d.

Graphing Exponential Functions Example 4 Graphing natural Exponential Functions Graph each natural exponential function a. b. Create a Table X -3 -2 -1 1 2 3 f(x) g(x) Graph of f(x) Graph of g(x)

Applications of Exponential Functions One of the most familiar examples of exponential growth is that of an investment earning continuously compounded interest. Formulas for Compound Interest After t years the balance A in an account with principal P and annual interest rate r (in decimal form) is given by the following formulas. 1. 2. Example 5 Compounding n times and Continuously A total of $12,000 is invested at an interest rate or 9%. Find the balance after 5 years if it is compounded quarterly and continuously. Quarterly: Formula: Substitution: Solution: Continuously: Formula: Substitution: Solution: Note that continuous compounding yields $93.64 more than quarterly.

Applications of Exponential Functions Example 6 Radioactive Decay In 1986, a nuclear reactor accident occurred in Chernobyl in what was then the Soviet Union. The explosion spread radioactive chemicals over hundreds of square miles, and the government evacuated the city and surrounding areas. To see why the city is now uninhabited, consider the following model. This model represents the amount of plutonium that remains of the initial 10 pounds after t years. How many grams of plutonium will remain after 100 years? How many grams of plutonium will remain after 10,000 years? From the graph you can see that Plutonium has a half life of 24,360 years.

Logarithmic Functions Previously we have studied functions and their inverses. During this study we realized that if a function has the property were no horizontal line intersect the graph more than once the function has an inverse. Looking at the graph of f(x) = ax we notice that f(x) has an inverse. Definition of Logarithmic Function For x>0 and a≠1, if and only if The function given by is called the logarithmic function with base a. When evaluating logarithms, remember that a logarithm is an exponent. This means that logax is the exponent to which a must be raised to obtain x. For instance, log28=3 because 2 raised to the 3 power is 8.

Evaluating Logarithms Example 7 Evaluating Logarithms Expression Value Justification a. b. c. d. e. f. 5 3 ½ -2 1

Evaluating Logarithms Example 7 Evaluating Logarithms on the Calculator Use the calculator to evaluate each expression. Expression Key Strokes Display a. b. c. LOG 10 ENTER 1 2 X LOG 2.5 ENTER 0.7958800 LOG (-) 2 ENTER ERROR Properties of Logarithms because then

Graphing Logarithmic Functions Example 8 Graphing a Logarithmic Function In the same coordinate plane graph the following two functions. a. b. x -2 -1 1 2 3 2x log10x

Natural Logarithmic Function The Natural Logarithmic Function The function defined by is called the natural logarithmic function. Properties of Natural Logarithms ln 1 = 0 ln e = 1 ln ex = x If ln x = ln y, then x = y Example 9 Using Properties of Natural Logarithms Example Solution Property a. b. c. d. Property 3 Property 1 Property 2

Domains of Logarithmic Functions Example 10 Finding the Domains of Logarithmic Functions Find the domain of each function a. f(x) = ln (x – 2) b. g(x) = ln (2 – x) c. Ln x2 Because ln (x – 2) is defined only if x – 2 > 0, it follows that the domain of f is (2, ∞). Because ln (2 – x) is defined only if 2 – x > 0, it follows that the domain of g is (- ∞, 2). Because ln x2 is defined only if x2 > , it follows that the domain of h is all real numbers except x = 0

Application of Natural Logarithms Example 11 Human Memory Models Students participating in a psychological experiment attended several lectures on a subject and were given an exam. Every month for a year after the exam, the students were retested to see how much of the material they remembered. The average scores for the group are given by the human memory model. where t is time in months. a. What was the original average score? c. What was the average score after six months? b. What was the average score after two months?

Change of Base Change of Base Formula Let a, b, and x be positive real numbers such that a ≠ 1 and b ≠ 1. Then Example 12 Changing Bases Using Common Logarithms Change each logarithmic function to base 10 and evaluate to nearest ten thousandth. Example Change of Base Evaluate a. b.

Properties of Logarithms Let a be a positive number such that a ≠ 1, and let n be a real number. If u and v are positive real numbers, the following properties are true. 1. 2. 3. 1. 2. 3. Example 13 Using the Properties of Logarithms Write the logarithm in terms of ln 2 and ln 3 a. b.

Using Properties of Logarithms Example 14 Using Properties of Logarithms Use the properties of logarithms to verify that Change one side to match the other. Original Statement Law of Negative Exponents ln un = n ln u Simplify Example 15 Rewrite Each Logarithm In Expanded Form Original Statement Product Rule Power Rule Original Statement Quotient Rule Power Rule

Using Properties of Logarithms Example 16 Rewrite each Logarithmic Expression in Condensed Form Original Statement Power Rule Of Logarithms Product Rule of Logarithms Example 17 Rewrite Each Logarithm In condensed Form Original Statement Product Rule Power Rule

Exponential and Logarithmic Equations Solving Exponential Equations Solving Logarithmic Equations Isolate the exponential expression Take the logarithm of both sides Solve for the variable Rewrite the equation in exponential form Solve for the variable Example 18 Solving an Exponential Equation Solve Original Equation Take logarithm of both sides Inverse Property Evaluate to the thousandths place Example 19 Solving an Exponential Equation Solve Original Equation Isolate the exponential expression Take the logarithm of both sides Inverse Property Evaluate to the thousandths place

Exponential and Logarithmic Equations Examples 20 & 21 Solving an Logarithmic Equation Solve Original Equation Solve Isolate the exponential expression Evaluate to nearest thousandths Take logarithm of both sides Inverse Property Solve Original Equation Quadratic Form Factor Set factors equal to zero Isolate exponential expression Take logarithm of both sides Evaluate to nearest thousandths

Exponential and Logarithmic Equations Example 20 Solving an Logarithmic Equation Solve Original Equation Isolate the Natural Logarithm Exponentiate both sides Inverse Property Solve Evaluate to the thousandths place Solve Original Equation Quotient Rule for Logarithms Exponentiate both sides Cross Multiply Subtract ex from both sides Factor Divide both sides by 1 - e

Applications Example 21 Doubling an Investment You have deposited $500 in an account that pays 6.75% interest, compounded continuously. How long will it take your money to double ? How long for it to triple? Double Investment Triple Investment Formula Formula Substitute Substitute Isolate Isolate Take Logarithm Take Logarithm Inverse Property Inverse Property Solve Solve Simplify Simplify

Application Example 22 Consumer Price Index for Sugar From 1970 to 1973, the consumer Price Index ( CPI ) value y for a fixed amount of sugar for the year t can be modeled by the equation where t = 10 represent 1970. During which year did the price of sugar reach four times its 1970 price of 30.5 on the CPI? Formula Substitute Isolate Take Logarithm Inverse Property Solve Since t = 0 represents 1970, the price of sugar reached 4 times its1079 price in 1988.