Copyright © 2011 Pearson Education, Inc. Exponential and Logarithmic Functions CHAPTER 12.1Composite and Inverse Functions 12.2Exponential Functions 12.3Logarithmic Functions 12.4Properties of Logarithms 12.5Common and Natural Logarithms 12.6Exponential and Logarithmic Equations with Applications 12

Copyright © 2011 Pearson Education, Inc. Composite and Inverse Functions Find the composition of two functions. 2.Show that two functions are inverses. 3.Show that a function is one-to-one. 4.Find the inverse of a function. 5.Graph a given function’s inverse function.

Slide Copyright © 2011 Pearson Education, Inc. Composition of functions: If f and g are functions, then the composition of f and g is defined as for all x in the domain f for which f(x) is the domain of g. The composition of g and f is defined as for all x in the domain of g for which f(x) is in the domain of g.

Slide Copyright © 2011 Pearson Education, Inc. If and find the following. Solution Example 1 In f(x), replace x with 1. Simplify. a. since g(3) = 2(3) – 5 = 1

Slide Copyright © 2011 Pearson Education, Inc. If and find the following. Solution continued In g(x), replace x with 17. Simplify. b.

Slide Copyright © 2011 Pearson Education, Inc. If and find the following. Solution continued Replace g(x) with 2x – 5. In f(x), replace x with 2x – 5. Simplify. c.

d. Slide Copyright © 2011 Pearson Education, Inc. continued Replace f(x) with 3x + 8. In g(x), replace x with 3x + 8 Simplify. d. If and find the following. Solution

Slide Copyright © 2011 Pearson Education, Inc. Inverse functions: Two functions f and g are inverses if and only if for all x in the domain of g and for all x in the domain of f. Inverse Functions To determine whether two functions f and g are inverses of each other, 1. Show that for all x in the domain of g. 2.Show that for all x in the domain of f.

Slide Copyright © 2011 Pearson Education, Inc. Solution Example 2a Verify that f and g are inverses. We need to show that and Since and, f and g are inverses.

Slide Copyright © 2011 Pearson Education, Inc. One-to-One function: A function f is one-to-one if for any two numbers a and b in its domain, when f(a) = f(b), a = b and when a b, f(a) f(b). Horizontal Line Test for One-to-One Functions Given a function’s graph, the function is one-to-one if every horizontal line that can intersect the graph does so at one and only one point.

Slide Copyright © 2011 Pearson Education, Inc. Determine whether the graph is a one-to-one function. Solution Example 3b A horizontal line can intersect the graph in more than one point, so the function is not one-to-one.

Slide Copyright © 2011 Pearson Education, Inc. Finding the Inverse Function of a One-to-One Function 1. If necessary, replace f(x) with y. 2.Replace all x’s with y’s and y’s with x’s. 3.Solve the equation from step 2 for y. 4.Replace y with f -1 (x). Existence of Inverse Functions A function has an inverse function if and only if the function is one-to-one.

Slide Copyright © 2011 Pearson Education, Inc. Solution Example 4a Find f -1 (x) for the function f(x) = 7x – 4. y = 7x – 4 x = 7y – 4 Since f[f -1 (x)]= x and f -1 [f (x)] = x, they are inverses.

Slide Copyright © 2011 Pearson Education, Inc. Graph of Inverse Functions The graphs of f and f -1 are symmetric with respect to the graph of y = x.

Slide Copyright © 2011 Pearson Education, Inc. Solution Example 5b Sketch the inverse of the function whose graph is shown. Draw the line y = x and reflect the graph on the line.

Slide Copyright © 2011 Pearson Education, Inc. If f(x) = x + 7 and g(x) = 2x – 12, what is a)  44 b)  3 c) 3 d) 44

Slide Copyright © 2011 Pearson Education, Inc. If f(x) = x + 7 and g(x) = 2x – 12, what is a)  44 b)  3 c) 3 d) 44

Slide Copyright © 2011 Pearson Education, Inc. Find f -1 (x) for 6x – 7. a) b) c) d)

Slide Copyright © 2011 Pearson Education, Inc. Find f -1 (x) for 6x – 7. a) b) c) d)

Copyright © 2011 Pearson Education, Inc. Exponential Functions Define and graph exponential functions. 2.Solve equations of the form b x = b y for x. 3.Use exponential functions to solve application problems.

Slide Copyright © 2011 Pearson Education, Inc. Exponential function: If b > 0, b 1, and x is any real number, then the exponential function is f(x) = b x. Note: The definition of the exponential function has two restrictions on b. If b = 1, then f(x) = b x = 1 x = 1, which is a linear function. If b < 0, then we could get values for which the function is not defined as a real number.

Slide Copyright © 2011 Pearson Education, Inc. Example 1a Graph f(x) = 2 x, and g(x) = 3 x.. Solution x–2–1012 f(x)f(x)124 g(x)g(x)139 Comparing the graphs, we can see the greater value of b, the steeper the graph. f(x) = 2 x g(x) = 3 x

Slide Copyright © 2011 Pearson Education, Inc. Example 2 Graph f(x) = 3 x+3. Solution x f(x)f(x) = = = = = 27 f(x) = 3 x+3

Slide Copyright © 2011 Pearson Education, Inc. Solving Exponential Equations 1. If necessary, write both sides of the equation as a power of the same base. 2.If necessary, simplify the exponents. 3.Set the exponents equal to each other. 4.Solve the resulting equation. The One-to-One Property of Exponentials Given b > 0 and b 1, if b x = b y, then x = y.

Slide Copyright © 2011 Pearson Education, Inc. Example 3 Solve. a. 3 x = 81 c. Solution a.3 x = 81 3 x = 3 4 x = 4 The solution set is {4}. c. The solution set is –4.

Slide Copyright © 2011 Pearson Education, Inc. Example 4 Rayanne deposited $15,000 in an account that pays 7% annual interest compounded quarterly. How much is accumulated in the account after 10 years? Solution Understand We are asked to find A and given that t = 10, P = $15,000, r = 0.07, and n = 4. Plan Use the formula

Slide Copyright © 2011 Pearson Education, Inc. continued Substitution. Simplify. Evaluate using a calculator and round to the nearest cent. Execute Answer After 10 years, the accumulated account is $30,

Slide Copyright © 2011 Pearson Education, Inc. continued Check Verify that the principal is $15,000 if the accumulated amount is $30, after the principal is compounded quarterly. Since the accumulated amount was rounded, it is expected our calculated value of the principal to be slightly different from $15,000.

Slide Copyright © 2011 Pearson Education, Inc. Half-Life The half-life of a radioactive substance is the amount of time it takes until only half of the original amount of the substance remains. The formula below gives the amount remaining, where A 0 is the initial amount, t is the time, and h is the half-life.

Slide Copyright © 2011 Pearson Education, Inc. Example 5 The isotope 45 Ca has a half-life of 165 days. How many grams of a 75-gram sample will remain after 1155 days? Solution Understand Given a 75-gram sample, we are to find the amount remaining after 1155 days. Plan Use the formula

Slide Copyright © 2011 Pearson Education, Inc. continued Substitute 75 for A, 1155 for t and 165 for h. Simplify. Evaluate using a calculator. Execute Answer 0.59 grams will remain after 1155 days. The check is left to the student.

Slide Copyright © 2011 Pearson Education, Inc. Which of the following functions does the graph represent? a) f(x) = 4 x+2 b) f(x) = 4 x–2 c) f(x) = 2 x+4 d) f(x) = 2 x–4

Slide Copyright © 2011 Pearson Education, Inc. Which of the following functions does the graph represent? a) f(x) = 4 x+2 b) f(x) = 4 x–2 c) f(x) = 2 x+4 d) f(x) = 2 x–4

Slide Copyright © 2011 Pearson Education, Inc. A stereo system is purchased for $2500. It’s value each year is about 85% of its value in the preceding year. Its value in dollars after t years is given by the exponential function V(t) = 2500(0.85) t. Find the salvage value of the stereo after 4 yr. a) $ b) $ c) $ d) $

Slide Copyright © 2011 Pearson Education, Inc. A stereo system is purchased for $2500. It’s value each year is about 85% of its value in the preceding year. Its value in dollars after t years is given by the exponential function V(t) = 2500(0.85) t. Find the salvage value of the stereo after 4 yr. a) $ b) $ c) $ d) $

Copyright © 2011 Pearson Education, Inc. Logarithmic Functions Convert between exponential and logarithmic forms. 2.Solve logarithmic equations by changing to exponential form. 3.Graph logarithmic functions. 4.Solve applications involving logarithms.

Slide Copyright © 2011 Pearson Education, Inc. Logarithm: If b > 0 and b 1, then y = log b x is equivalent to x = b y. x = b y y = log b x The exponent is the logarithm. The base is the base of the logarithm.

Slide Copyright © 2011 Pearson Education, Inc. Example 1 Write in logarithmic form. a.b.c. Solution a.b.c.

Slide Copyright © 2011 Pearson Education, Inc. Example 2 Write in exponential form. a.b.c. Solution a.b.c.

Slide Copyright © 2011 Pearson Education, Inc. Solving Logarithmic Equations To solve an equation of the form log b x = y, where b, x, or y is a variable, write the equation in exponential form, b y = x, and then solve for the variable.

Slide Copyright © 2011 Pearson Education, Inc. Example 3 Solve. a.b. Solution a.b.

Slide Copyright © 2011 Pearson Education, Inc. continued Solve. c.d. Solution c.d.

Slide Copyright © 2011 Pearson Education, Inc. Example 4 Solve Solution Begin by rewriting the equation in exponential form and then solve. Divide both sides by -3 to isolate the logarithm. Simplify. Subtract 11 from both sides. Write in exponential form.

Slide Copyright © 2011 Pearson Education, Inc. For any real number b, where b > 0 and b ≠ 1, 1. log b b = 1 2. log b 1 = 0

Slide Copyright © 2011 Pearson Education, Inc. Example 5 Find the value. a.b. Solution a.b.

Slide Copyright © 2011 Pearson Education, Inc. Graphing Logarithmic Functions To graph a function in the form f(x) = log b x, 1. Replace f(x) with y and write the logarithm in exponential form x = b y. 2. Find ordered pairs that satisfy the equation by assigning values to y and finding x. 3.Plot the ordered pairs and draw a smooth curve through the points.

Slide Copyright © 2011 Pearson Education, Inc. Example 6b Graph f(x) = log 1/4 x. Solution Replacing f(x) by y, we see that the equation y = log 1/4 x can be rewritten in exponential form as y –2 – x = 0

Slide Copyright © 2011 Pearson Education, Inc. Example 7 The function P = 95 – 30 log 2 x models the percent, P, of students who recall the important features of a classroom lecture over time, where x is the number of days that have elapsed since the lecture was given. What percent of the students recall the important features of a lecture 8 days after it was given? (Source: Psychology for the New Millennium, 8 th Edition, Spencer A. Rathos, Thomson Publishing Company) Solution Understand We are given the function that models the percent, P, of students who recall the important features of a lecture x days after it is given. We are to find the percent of students who recall the important features of a lecture 8 days after it was given.

Slide Copyright © 2011 Pearson Education, Inc. continued Execute Answer 5% of the students remember the important features of a lecture 8 days after it is given. Plan Evaluate P = 95 – 30log 2 x where x = 8. P = 95 – 30log 2 8 P = 95 – 30(3) P = 95 – 90 P = 5 Check 5 = 95 – 30log 2 x –90 = – 30log 2 x 3 = log 2 x 2 3 = x 8 = x

Slide Copyright © 2011 Pearson Education, Inc. Write 6 2 = 36 in logarithmic form. a) b) c) d)

Slide Copyright © 2011 Pearson Education, Inc. Write 6 2 = 36 in logarithmic form. a) b) c) d)

Slide Copyright © 2011 Pearson Education, Inc. Write in exponential form. a) b) c) d)

Slide Copyright © 2011 Pearson Education, Inc. Write in exponential form. a) b) c) d)

Slide Copyright © 2011 Pearson Education, Inc. Solve log 3 x = 5. a)  243 b)  125 c) 125 d) 243

Slide Copyright © 2011 Pearson Education, Inc. Solve log 3 x = 5. a)  243 b)  125 c) 125 d) 243

Copyright © 2011 Pearson Education, Inc. Properties of Logarithms Apply the inverse properties of logarithms. 2.Apply the product, quotient, and power properties of logarithms.

Slide Copyright © 2011 Pearson Education, Inc. Inverse Properties of Logarithms For any real numbers b and x, where b > 0 and b 1, and x > 0, 1. 2.

Slide Copyright © 2011 Pearson Education, Inc. Example 1 Find the value. a. b. Solution a. b.b.

Slide Copyright © 2011 Pearson Education, Inc. continued Find the value. c. d. Solution a. b.b.

Slide Copyright © 2011 Pearson Education, Inc. Further Properties of Logarithms For real numbers x, y, and b, where x > 0, y > 0, b > 0, and b ≠ 1. Product Rule of Logarithms: log b xy = log b x + log b y Quotient Rule of Logarithms: Power Rule of Logarithms: log b x r = rlog b x (The logarithm of the product of two numbers is equal to the sum of the logarithms of the numbers.) (The logarithm of the quotient of two numbers is equal to the difference of the logarithms of the numbers.) (The logarithm of a number raised to a power is equal to the exponent times the logarithm of the number.)

Slide Copyright © 2011 Pearson Education, Inc. Example 2 Use the product rule of logarithms to write each expression as a sum of logarithms. a. log b pqrb. log b x(x + 5) Solution a. log b pqr = log b p + log b q + log b r b. log b x(x+5) = log b x + log b (x + 5)

Slide Copyright © 2011 Pearson Education, Inc. Example 3 Use the product rule of logarithms in the form log b xy = log b x + log b y to write the expression as a single logarithm. a. b. Solution a. b.b.

Slide Copyright © 2011 Pearson Education, Inc. Example 4 Use the quotient rule of logarithms to write the expression as a difference of logarithms. Leave the answers in simplest form. a. b. Solution a. b.b.

Slide Copyright © 2011 Pearson Education, Inc. Example 5a Use the quotient rule of logarithms in the form to write the expression as a single logarithm. Solution

Slide Copyright © 2011 Pearson Education, Inc. Example 5b Use the quotient rule of logarithms in the form to write the expression as a single logarithm. Solution

Slide Copyright © 2011 Pearson Education, Inc. Example 6 Use the power rule of logarithms to write the expression as a multiple of a logarithm. Solution a. log 3 x 11 log 3 x 11 = 11 log 3 x b. a. b. = log b x -7 = -7log b x

Slide Copyright © 2011 Pearson Education, Inc. Example 7 Use the power rule of logarithms to write the expression as a logarithm of a quantity to a power. Leave the answers in simplest form without negative or fractional exponents. Solution a. b. a. b.

Slide Copyright © 2011 Pearson Education, Inc. continued Use the power rule of logarithms to write the expression as a logarithm of a quantity to a power. Leave the answers in simplest form without negative or fractional exponents. Solution c.

Slide Copyright © 2011 Pearson Education, Inc. Example 8a Write the expression as a sum or difference of multiples of logarithms. Solution Use the product and quotient rules. = log b x 7 + log b z 9  log b y 4 Use the power rule. = 7 log b x + 9 log b z  4 log b y

Slide Copyright © 2011 Pearson Education, Inc. Example 8b Write the expression as a sum or difference of multiples of logarithms. Solution

Slide Copyright © 2011 Pearson Education, Inc. Example 8c Write the expression as a sum or difference of multiples of logarithms. Solution

Slide Copyright © 2011 Pearson Education, Inc. Example 9a Write the expression as a single logarithm. Leave the answers in simplest form without negative or fractional exponents. Solution

Slide Copyright © 2011 Pearson Education, Inc. Example 9b Write the expression as a single logarithm. Leave the answers in simplest form without negative or fractional exponents. Solution

Slide Copyright © 2011 Pearson Education, Inc. Example 9c Write the expression as a single logarithm. Leave answer in simplest form without negative or fractional exponents. Solution

Slide Copyright © 2011 Pearson Education, Inc. Use the power rule to write the expression as a multiple of a logarithm. a) b) c) d)

Slide Copyright © 2011 Pearson Education, Inc. Use the power rule to write the expression as a multiple of a logarithm. a) b) c) d)

Slide Copyright © 2011 Pearson Education, Inc. Express as a single logarithm: a) b) c) d)

Slide Copyright © 2011 Pearson Education, Inc. Express as a single logarithm: a) b) c) d)

Copyright © 2011 Pearson Education, Inc. Common and Natural Logarithms Define common logarithms and evaluate them using a calculator. 2.Solve applications using common logarithms. 3.Define natural logarithms and evaluate them using a calculator. 4.Solve applications using natural logarithms.

Slide Copyright © 2011 Pearson Education, Inc. Common logarithms: Logarithms with a base of 10. Log 10 x is written as log x. Note that log 10 = 1. Solution Example 1 Use a calculator to approximate each common logarithm. Round to the nearest thousandth if necessary. a. log 456b. log a. log 456 b.log

Slide Copyright © 2011 Pearson Education, Inc. Example 2 The function is used to calculate sound intensity, where d represents the intensity in decibels, I represents the intensity in watts per unit of area, and I 0 represents the faintest audible sound to the average human ear, which is watts per square meter. What is the intensity level of sounds at a decibel level of 75 dB? Understand We are given the function, and we are to find the sound intensity, I.

Slide Copyright © 2011 Pearson Education, Inc. continued Execute Answer The sound intensity is 10 –4.5. Plan Using, substitute 75 for d and for I 0 and then solve for I.

Slide Copyright © 2011 Pearson Education, Inc. continued Check If the sound intensity is , verify the decibel reading is 75.

Slide Copyright © 2011 Pearson Education, Inc. Natural logarithms: Base-e logarithms are called natural logarithms and log e x is written as ln x. Note that ln e = 1. Solution Example 3 Use a calculator to approximate each natural logarithm to four decimal places. a. ln 67 b. ln a. ln 67 b.ln

Slide Copyright © 2011 Pearson Education, Inc. Example 4 If an account pays 8% annual interest, compounded continuously, how long will it take a deposit of $25,000 to produce an account balance of $100,000? Understand We are to find the time it takes for $25,000 to grow to $100,000 if it is compounded continuously at 8%. Plan In, replace P with 25,000, r with 0.08, A with $100,000, and then simplify.

Slide Copyright © 2011 Pearson Education, Inc. continued Answer The account balance will reach $100,000 in approximately years. Substitute. Divide. Approximate using a calculator. Execute

Slide Copyright © 2011 Pearson Education, Inc. continued Check Because was not the exact time, $100, is reasonable.

Slide Copyright © 2011 Pearson Education, Inc. Use a calculator to approximate log to four decimal places. a)  b)  c) d)

Slide Copyright © 2011 Pearson Education, Inc. Use a calculator to approximate log to four decimal places. a)  b)  c) d)

Slide Copyright © 2011 Pearson Education, Inc. Use a calculator to approximate ln 21 to four decimal places. a) b) c) d)

Slide Copyright © 2011 Pearson Education, Inc. Use a calculator to approximate ln 21 to four decimal places. a) b) c) d)

Copyright © 2011 Pearson Education, Inc. Exponential and Logarithmic Equations with Applications Solve equations that have variables as exponents. 2.Solve equations containing logarithms. 3.Solve applications involving exponential and logarithmic functions. 4.Use the change-of-base formula.

Slide Copyright © 2011 Pearson Education, Inc. Properties for Solving Exponential and Logarithmic Equations For any real numbers b, x, and y, where b > 0 and b 1, 1. If b x = b y, then x = y. 2. If x = y, then b x = b y. 3.For x > 0 and y > 0, if log b x = log b y, then x = y. 4.For x > 0 and y > 0, if x = y, then log b x = log b y. 5.For x > 0, if log b x = y, then b y = x.

Slide Copyright © 2011 Pearson Education, Inc. Example 1 Solve 8 x = 15. Solution Use if x = y, then log b x = log b y (property 4). Divide both sides by log 8. The exact solution is. Using a calculator, we find. Check8 x = = = 15 The answer is correct.

Slide Copyright © 2011 Pearson Education, Inc. Example 2 Solve e 2x = 12. Solution

Slide Copyright © 2011 Pearson Education, Inc. Solving Equations Containing Logarithms To solve equations containing logarithms, use the properties of logarithms to simplify each side of the equation and then use one of the following. If the simplification results in an equation in the form log b x = log b y, use the fact that x = y, and then solve for the variable. If the simplification results in an equation in the form log b x = y, write the equation in exponential form, b y = x, and then solve for the variable (as we did in Section 10.3).

Slide Copyright © 2011 Pearson Education, Inc. Example 3a Solve Solution Logarithms are defined for positive numbers only. A check will show that x = 5 the only solution.

Slide Copyright © 2011 Pearson Education, Inc. Example 3b Solve Solution This equation is in the form log b x = y, so write it in exponential form, b y = x. Solve for x.

Slide Copyright © 2011 Pearson Education, Inc. Example 3c Solve. Solution These 2 solutions check and are the solutions for this equation.

Slide Copyright © 2011 Pearson Education, Inc. Example 4 If $1000 is deposited into an account at 7.6% interest compounded continuously, how much money will be in the account after 8 years? Understand We are given P = $1000, r = 0.076, and t = 8 and we are asked to find A. Plan Use Execute

Slide Copyright © 2011 Pearson Education, Inc. continued There will be $ in the account after 8 years. Answer CheckUse the formula It checks.

Slide Copyright © 2011 Pearson Education, Inc. Example 7 A nuclear reactor contains 200 grams of radioactive plutonium 239 P. Plutonium disintegrates according to the formula A = A 0 e t. How much will remain after 5000 years? Understand We are given A 0 = 200 g and t = 5000 years and are asked to find A. Plan Use Execute Simplify. Substitute.

Slide Copyright © 2011 Pearson Education, Inc. continued About grams will remain after 5000 years. Answer Check Use A = A 0 e t to verify that it takes 5000 years for 200 grams of 239 P to disintegrate to grams. It checks = 200e t

Slide Copyright © 2011 Pearson Education, Inc. Change-of-Base Formula In general, if a > 0, a 1, b > 0, b 1, and x > 0, then In terms of common and natural logarithms,

Slide Copyright © 2011 Pearson Education, Inc. Example 8 Use the change-of-base formula to calculate log Round the answer to four decimal places. Solution Check The answer is correct.

Slide Copyright © 2011 Pearson Education, Inc. Solve log x + log(x – 3) = 1. a) x =  2 b) x = 5 c) x =  2 or x = 5 d) x is undefined.

Slide Copyright © 2011 Pearson Education, Inc. Solve log x + log(x – 3) = 1. a) x =  2 b) x = 5 c) x =  2 or x = 5 d) x is undefined.

Slide Copyright © 2011 Pearson Education, Inc. If $500 is deposited into an account at 9% interest compounded continuously, how much will be in the account after 5 years? a) $ b) $ c) $ d) $45,008.57

Slide Copyright © 2011 Pearson Education, Inc. If $500 is deposited into an account at 9% interest compounded continuously, how much will be in the account after 5 years? a) $ b) $ c) $ d) $45,008.57

Slide Copyright © 2011 Pearson Education, Inc. Use the change-of-base formula to approximate log a) 0.48 b) 0.85 c) 2.07 d) 8

Slide Copyright © 2011 Pearson Education, Inc. Use the change-of-base formula to approximate log a) 0.48 b) 0.85 c) 2.07 d) 8