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1 Chapter 13 Exponential and Logarithmic Functions

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2 Definition of an Exponential Function The exponential function with base b is denoted by So, in an exponential function, the variable is in the exponent. Section 13.1: Exponential Functions and Their Graphs

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3 Exponential Functions Which of the following are exponential functions?

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4 Graphs of Exponential Functions They can be broken into two categories— exponential growth, and exponential decay (decline).

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5 The Graph of an Exponential Growth Function We will look at the graph of an exponential function that increases as x increases, known as the exponential growth function. It has the form Example: y = 2 x Notice the rapid increase in the graph as x increases The graph increases slowly for x < 0. y-intercept is (0, 1) Horizontal asymptote is y = 0. xy y = 2 x

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7 The Graph of an Exponential Decay (Decline) Function We will look at the graph of an exponential function that decreases as x increases, known as the exponential decay function. It has the form Example: y = 2 -x Notice the rapid decline in the graph for x < 0. The graph decreases more slowly as x increases. y-intercept is (0, 1) Horizontal asymptote is y = 0. xy y = 2 -x

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8 Graphs of Exponential Functions Notice that f(x) = 2 x and g(x) = 2 -x are reflections of one another about the y-axis. Both graphs have y-intercept ___________ and horizontal asymptote ________. The domain of f(x) and g(x) is _________; the range is _______.

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9 Graphs of Exponential Functions Also, note that, applying the properties of exponents. So an exponential function is a decay function if The base b is greater than one and the function is written as f(x) = b -x -OR- The base b is between 0 and 1 and the function is written as f(x) = b x

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10 Graphs of Exponential Functions Examples: In this case, b = 0.25 (0 < b < 1). In this case, b = 5.6 (b > 1).

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11 Natural base e It may seem hard to believe, but when working with exponents and logarithms, it is often convenient to use the irrational number e as a base. The number e is defined as This value approaches as x approaches infinity.

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12 Evaluating the Natural Exponential Function To evaluate the function f(x) = e x, we will use our calculators to find an approximation. You should see the e x button on your graphing calculator (Use ). Example: Given, find f(3) and f(-0.5) to 3 decimal places. ≈ ____________ ≈ _______________

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13 Graphing the Natural Exponential Function Growth or decay? Domain: Range: Asymptote: x-intercept: y-intercept: List four points that are on the graph of f(x) = e x.

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14 Graphing the Natural Exponential Function Determine the following: Growth or decay? Domain: Range: Asymptote: x-intercept: y-intercept:

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15 Example The population of a town is modeled by the function where t = 0 corresponds to 1990 and P is the town’s population in thousands. a) According to the model, what was the town’s population in 1990? b) According to the model, what was the town’s population in 2008?

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16 Example (continued) c) Graph the function on your calculator and determine in which year the town’s population reached 75,000 people. How would we solve this algebraically??

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17 Now that you have studied the exponential function, it is time to take a look at its INVERSE: the LOGARITHMIC FUNCTION. In the exponential function, the independent variable was the exponent. So we substituted values into the exponent and evaluated it for a given base. For example, for f(x) = 2 x f(3) = Section 13.2: Logarithmic Functions

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18 Logarithmic Functions For the inverse (logarithmic) function, the base is given and the answer is given, so to evaluate a logarithmic function is to find the exponent. That is why I think of the logarithmic function as the “Guess That Exponent” function. Warm Up: Give the value of ? in each of the following equations.

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19 Logarithmic Functions (continued) For example, to evaluate log 2 8 means to find the exponent such that 2 raised to that power gives you 8.

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20 The following definition demonstrates this connection between the exponential and the logarithmic function. Definition of an Logarithmic Function For y > 0, b > 0, and b ≠ 1, If y = b x, then x = log b y y = b x is the exponential form x = log b y is the logarithmic form We read log b y as “log base b of y”. Logarithmic Functions (continued)

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21 Subliminal Message: The exponential and logarithmic functions of the same base are inverses.

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22 Converting Between Exponential and Logarithmic Forms I. Write the logarithmic equation in exponential form. a) b) II. Write the exponential equation in logarithmic form. a) b) If y = b x, then x = log b y

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23 Evaluating Logarithms w/o a Calculator To evaluate logarithmic expressions by hand, we can use the related exponential expression. Example: Evaluate the following logarithms:

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24 Evaluating Logarithms w/o a Calculator (cont.)

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25 Evaluating Logarithms w/o a Calculator Okay, try these. e) f) g) h)

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26 Determine the value of the unknowns a) b)

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27 Determine the value of the unknowns c) d)

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28 Graphs of Logarithmic Functions Example: Graph f(x) = 2 x and g(x) = log 2 x in the same coordinate plane. Solution: To do this, make a table of values for f(x) and then switch the x and y coordinates to make a table of values for g(x).

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29 Graphs of Logarithmic Functions (continued) f(x) = 2 x g(x) = log 2 x xf(x) -41/16 -21/ xg(x) 1/16-4 1/ f(x) = 2 x g(x)= log 2 x y =x Inverse functions

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30 Graphs of Logarithmic Functions (continued) Notice how the domain and range of the inverse functions are switched. The exponential function has Domain: ____________ Range: ____________ Horizontal asymptote: _________ The logarithmic function has Domain: __________ Range: ___________ Vertical asymptote: __________ f(x) = 2 x g(x)= log 2 x y =x

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31 Not all logarithmic expressions can be evaluated easily by hand. In fact, most cannot. For example, to evaluate is to find x such that 2 x = 175. This is not a simple task. In fact, the answer is irrational. For these types of problems, we will use the calculator. “Calculators?? Back in my day, we used log tables and slide rules!” Section 13.4: Evaluating Common Logarithms with a Calculator

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32 The calculator, however, only calculates two different base logarithms—the common logarithm and the natural logarithm. I. The COMMON LOGARITHM is the logarithmic function with base 10. On the TI-83/84, look for the button. This is used to evaluate the common log (base 10) only. Example: Evaluate f(x)=log 10 x for x = 400. Round to four decimal places. Solution: f(400) = log Answer: ___________ Evaluating Common Logarithms with a Calculator (continued) LOG ENTER

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33 We can also find a number given its logarithm. We say that N is the antilog of We use 2 ND LOG [10 x ] Example: log N = N = _____________________________ Antilog of the Common Log

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34 Application of the Logarithm Example Measured on the Richter, the magnitude of an earthquake of intensity I is defined to be R = Log(I/I 0 ), where I 0 is a minimum level for comparison. What is the Richter scale reading for the 1995 Philippine earthquake for which I=20,000,000 I 0 ?

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35 In section 13.1, we saw the natural exponential function with base e. Its inverse is the natural logarithmic function with base e. Instead of writing the natural log as log e x, we use the notation ln x, which is read as “the natural log of x” and is understood to have base e. Section 13.5: The Natural Logarithmic Function

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36 The Natural Logarithmic Function To evaluate the natural log using the TI-83/84, use the button. Example Evaluate the function f(x) = ln x at a) x = 1.5 b) x = -2.3 LN This means that ______________________________________

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37 Graph of the Natural Exponential and Natural Logarithmic Function f(x) = e x and g(x) = ln x are inverse functions and, as such, their graphs are reflections of one another in the line y = x.

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38 We say that N is the antilog of We use 2 ND LOG [e x ] Example: ln N = N = _____________________________ Antilog of the Natural Logarithm LN2ND

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39 Change-of-Base Formula I mentioned that the calculator only has two types of log keys, the COMMON LOG (BASE 10) and the NATURAL LOG (BASE e). It’s true that these two types of logarithms are used most often, but sometimes we need to evaluate logarithms with bases other than 10 or e. To do this on the calculator, we use a CHANGE-OF-BASE FORMULA. We will convert the logarithm with base a into an equivalent expression involving common logarithms or natural logarithms.

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40 Change-of-Base Formula (continued) Change-of-Base Formula Let a, b, and x be positive real numbers such that a 1 and b 1. Then log b x can be converted to a different base using any of the following formulas.

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41 Change-of-Base Formula Examples* Example: Use the change-of-base formula to evaluate log a) using common logarithms b) using natural logarithms. Solution: The result is the same whether you use the common log or the natural log.

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42 Change-of-Base Formula Examples Example Use the change-of-base formula to evaluate a) b)

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43 a) Graph on the calculator. b) Graph its inverse on the calculator. Graph of the Logarithmic Function with base b

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44 Let b be a positive real number such that b 1, and let n, x, and y be real numbers. Base b Logarithms Natural Logarithms Section 13.3: Properties of Logarithms

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45 WARNING!!!!!!

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46 Use the properties of logs to EXPAND each of the following expressions into a sum, difference, or multiple of logarithms:

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47 Again! Use the properties of logs to EXPAND each of the following expressions into a sum, difference, or multiple of logarithms:

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48 This is fun! Use the properties of logs to EXPAND each of the following expressions into a sum, difference, or multiple of logarithms:

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49 Try these! Use the properties of logs to CONDENSE each of the expressions into a logarithm of a single quantity:

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50 Properties of Logarithms Rock! Use the properties of logs to CONDENSE each of the expressions into a logarithm of a single quantity:

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51 One more! Use the properties of logs to CONDENSE each of the expressions into a logarithm of a single quantity:

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52 Solving EXPONENTIAL Equations: Part I I. Using the One-to-One Property If you can write the equation so that both sides are expressed as powers of the SAME BASE, you can use the property b x = b y if and only if x = y. Example: Solve 4 x-2 = 64 Section 13.6: Solving Exponential and Logarithmic Equations

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53 Solving EXPONENTIAL Equations: Part II II. By Taking the Logarithm of Each Side 1. ISOLATE the exponential term on one side of the equation. 2. TAKE THE COMMON OR NATURAL LOG of each side of the equation. 3. USE THE PROPERTIES OF LOGARITHMS to remove the variable from the exponent. 4. SOLVE for the variable. Use the calculator to evaluate the resulting log expression.

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54 Example: S olve 3(5 4x+1 ) -7 = 10 Give answer to 3 decimal places. Solving EXPONENTIAL Equations

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55 Solving LOGARITHMIC Equations: Part I I. Using the One-to-One Property If you can write the equation so that both sides are expressed as SINGLE logarithms with the SAME BASE, you can use the property log b x = log b y if and only if x = y.

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56 Solving LOGARITHMIC Equations: Part I Example of one-to-one property: Solve log 3 x + 2log 3 5 = log 3 (x + 8)

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57 Solving Logarithmic Equations: Part II II. By Rewriting in Exponential Form 1. USE THE PROPERTIES OF LOGARITHMS to combine log expressions into a SINGLE log expression, if necessary. 2. ISOLATE the logarithmic expression on one side of the equation. 3. Rewrite the equation in EXPONENTIAL FORM. 4. SOLVE the resulting equation for the variable. 5. CHECK the solution in the original equation either graphically or algebraically

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58 Example: Solve Solving Logarithmic Equations

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59 Solving Exponential and Logarithmic Equations GRAPHICALLY Remember, you can verify the solution of any one of these equations by finding the graphical solution using your TI-83/84 calculator. Enter the left hand side of the original equation as y 1, Enter the right side as y 2, and Find the point at which the graphs intersect. Below is the graphical solution for the last example. The x-coordinate of the intersection point is approximately , confirms our algebraic solution.

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60 I. Solve each of the following EXPONENTIAL equations. Round to 4 decimal places, if necessary.

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67 Challenge Question

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68 II. Solve each of the following LOGARITHMIC equations. Round to 4 decimal places, if necessary.

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71 Applications Example: How long will it take $25,000 to grow to $500,000 if it is invested at 9% annual interest compounded monthly? Round to the nearest tenth of a year. Formula:

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72 Another Example: The population of Asymptopia was 6500 in 1985 and has been tripling every 12 years since then. If this rate continues, when will the population reach 75,000? Let t represent the number of years since 1985 P(t) represents the population after t years.

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73 Drug medication: The formula can be used to find the number of milligrams D of a certain drug that is in a patient’s bloodstream h hours after the drug has been administered. When the number of milligrams reaches 2, the drug is to be administered again. What is the time between injections? Round to the nearest tenth of an hour.

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74 A Logarithmic Model: The loudness L, in bels (named after ?), of a sound of intensity I is defined to be where I 0 is the minimum intensity detectable by the human ear. The bell is a large unit, so a subunit, the decibel, is generally used. For L, in decibels, the formula is

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75 A Log Model (cont) Find the loudness, in decibels, for each sound with the given intensity. a) Library b) Dishwasher c) Loud muffler

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76 A Log Model (cont) If the front rows of a rock concert has a loudness of 110 dB and normal conversation has a loudness of 60 dB, how many times greater is the intensity of the sound in the front rows of a rock concert than the intensity of the sound of normal conversation?

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77 Let’s say we want to plot the graph y = 5 x xy The detail for values of x less than 3 is nearly imperceptible. Section 13.7: Graphs on Log and Semi-log Paper

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78 Often times, we want to model data that require that small variations at one end of the scale are visible, while large variations at the other end are also visible. To graph functions where one or both of the variables have a wide change in values, we can use a logarithmic scale. This type of scale is marked off in distances that are proportional to the logarithm of the values being represented. The distances between integers on a log scale are not equal, but will give us a better way to show a greater range of values.

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79 If we want to show a large range of values for only one of your variables, we will use SEMI-LOG paper. Semi-log paper has two scales: The horizontal scale has equal spacing between the lines The vertical scale does not have equal spacing between the lines. It uses a logarithmic scale.

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80 Semi-log paper allows you to graph exponential data without having to translate your data into logarithms—the paper does it for you. The scale of semi-log paper has cycles. Below is what is known as 3- cycle semi-log graph paper. On the vertical scale, the powers of ten are evenly spaced. On the horizontal scale, the numbers along the axis are evenly spaced.

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81 Let’s see the graph of y = 5 x on semi-log paper. xy

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82 Semi-log paper is often used to transform a nonlinear data relation into a linear one. If a function makes a STRAIGHT LINE when graphed on semi-log graph paper, we call it an EXPONENTIAL FUNCTION.

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83 On log-log paper, both axes are marked with a logarithmic scale.

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84 Example: Create a log-log plot of the function y = 0.5x 3. xy

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85 Notice that the graph of y = 0.5x 3 on log-log paper is a straight line. An equation in the form of y = ax b is called a POWER FUNCTION. If you plot the data points of a power function on log-log paper, it appears LINEAR.

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