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Unformatted text preview: MAC 1105 Module 8 Exponential and Logarithmic Functions I Rev.S08 Learning Objectives Upon completing this module, you should be able to: 1. 2. 3. 4. 5. 6. 7. Distinguish between linear and exponential growth. Model data with exponential functions. Calculate compound interest. Use the natural exponential function in applications. Evaluate the common logarithmic function. Evaluate the natural logarithmic function. Solve basic exponential and logarithmic equations. Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 2 Exponential and Logarithmic Functions I There are two major topics in this module: - Rev.S08 Exponential Functions Logarithmic Functions http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 3 1 A Quick Review on Function y = f(x) means that given an input x, there is means jjust one corresponding output y. ust Graphically, this means that the graph passes the vertical line test. passes vertical Numerically, this means that in a table of values for y = f(x) there are no x -values values there repeated. Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 4 Example Given y2 = x, iis y = f(x)? That is, is y a function s Given )? of x? of No, because if x = 4, y could be 2 or –2. No, Note that the graph fails the vertical line test. x y 4 –2 1 –1 0 0 1 1 4 2 Rev.S08 Note that there is a value of x in the table for which Note there are two different values of y (that is, x -values there are repeated.) http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 5 What is One-to-One? • Given a function y = f(x), f is one-to-one means that given an output y there was just one input x which produced that output. • Graphically, this means that the graph passes the horizontal line test. Every horizontal line intersects the graph at most once. • Numerically, this means the there are no y values repeated in a table of values. Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 6 2 Example Given y = f(x) = |x|, is f one-to-one? Given |, one-to-one No, because if y = 2, x could be 2 or – 2. No, Note that the graph fails the horizontal line test. Note horizontal x y –2 2 –1 0 1 1 2 2 (2,2) 1 0 Rev.S08 (-2,2) Note that there is a value of y in the table for which Note there are two different values of x (that is, ythere values are repeated.) http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 7 What is the Definition of a One-to-One Function? A function f is a one-to-one function if, for elements c o ne-to-one and d in the domain of f , and c d implies f (c) f (d ) Example: Given y = f ( x) = |x| , f is not one-to-one Example: |, is one-to-one because –2 2 yet | – 2 | = | 2 | because Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 8 What is an Inverse Function? -1 • f -1 is a symbol for the inverse of the function f , not to inverse be confused with the reciprocal. • If f -1(x ) does NOT mean 1/ f (x), what does it mean? If -1 does NOT -1 • y = f -1(x ) means that x = f (y ) means -1 • Note that y = f -1( x) iis pronounced “ y equals f s Note iinverse of x. ” nverse Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 9 3 Example of an Inverse Function Let F be Fahrenheit temperature and let C be Centigrade temperature. F = f(C) = (9/5)C + 32 -1 C = f -1(F) = ????? The function f multiplies an input C by 9/5 and adds 32. The To undo multiplying by 9/5 and adding 32, one should subtract 32 and divide by 9/5 -1 So C = f -1(F) = (F – 32)/(9/5) So -1 C = f -1(F ) = (5/9)(F – 32) Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 10 Example of an Inverse Function (Cont.) F = f(C) = (9/5)C + 32 C = f -1(F) = (5/9)(F – 32) Evaluate f(0) and interpret. f(0) = (9/5)(0) + 32 = 32 When the Centigrade temperature is 0, the Fahrenheit temperature is 32. Evaluate f -1(32) and interpret. f -1(32) = (5/9)(32 - 32) = 0 When the Fahrenheit temperature is 32, the Centigrade temperature is 0. Note that f(0) = 32 and f -1(32) = 0 Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 11 Graph of Functions and Their Inverses -1 The graph of f -1 is a reflection of the graph of The reflection f across the line y = x Note that the domain of f equals the r ange of f Note -1 range of f equals the domain of f -1 . range Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. -1 -1 and the 12 4 How to Find Inverse Function Symbolically? -1 Check that f is a one-to-one function. If not, f -1 does Check one-to-one If not exist. Solve the equation y = f(x) for x, resulting in the Solve for -1 equation x = f -1(y) equation -1 Interchange x and y to obtain y = f -1(x) Interchange Example. f( x ) = 3 x + 2 y = 3x + 2 Solving for x gives: 3x = y – 2 Solving x = (y – 2)/3 Interchanging x and y g ives: y = (x – 2)/3 Interchanging -1 -1( x ) = (x – 2)/3 So f So http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08 13 How to Evaluate Inverse Function Numerically? x 1 2 3 4 5 f(x) –5 –3 0 3 5 -1 The function is one-to-one, so f -1 The one-to-one exists. -1 f -1 ( –5) = 1 -1 f -1 ( –3) = 2 -1 f -1 ( 0) = 3 -1 f -1 ( 3) = 4 -1 f -1 ( 5) = 5 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08 14 How to Evaluate Inverse Function Graphically? The graph of f below passes the horizontal passes horizontal line test so f is one-too ne-toone. Evaluate f -1(4). Evaluate -1 Since the point (2,4) is on the graph of f , the point the (4,2) will be on the graph -1 of f -1 and thus f -1(4) = 2. of -1 Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. f(2)=4 15 5 What is the Formal Definition of Inverse Functions? -1 L et f be a one-to-one function. Then f -1 is Let the inverse function of f, if the -1 -1 • (f -1 o f)(x) = f -1(f(x)) = x for every x in the )) domain of f domain -1 -1 )) • (f o f -1 )(x) = f(f -1(x)) = x for every x in the -1 domain of f -1 domain Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 16 Exponential Functions and Models We will start with population growth. Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 17 Population Growth Suppose a population is 10,000 in January 2004. Suppose the population increases by… 500 people per year What is the population in Jan 2005? 10,000 + 500 = 10,500 10,000 10,500 What is the population in Jan 2006? 10,500 + 500 = 11,000 10,500 11,000 Rev.S08 5% per year What is the population in Jan 2005? 10,000 + .05(10,000) = 10,000 + 500 = 10,500 10,000 10,500 What is the population in Jan 2006? 10,500 + .05(10,500) = 10,500 + 525 = 11,025 10,500 11,025 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 18 6 Population Growth (Cont.) Suppose a population is 10,000 in Jan 2004. Suppose the population increases by 500 per year. What is the population by population in … . population Jan 2005? 10,000 + 500 = 10,500 Jan 2006? 10,000 + 2(500) = 11,000 Jan 2007? 10,000 + 3(500) = 11,500 Jan 2008? 10,000 + 4(500) = 12,000 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08 19 Population Growth (Cont.) Suppose a population is 10,000 in Jan 2004 and increases Suppose i ncreases by 500 per year. Let t be the number of years after 2004. Let P(t) be the Let population in year t . What is the symbolic population What representation for P(t )? We know… representation Population in 2004 = P( 0 ) = 10,000 + 0 (500) Population 10,000 Population in 2005 = P( 1 ) = 10,000 + 1 (500) Population 10,000 Population in 2006 = P( 2 ) = 10,000 + 2 (500) Population 10,000 Population in 2007 = P( 3 ) = 10,000 + 3 (500) Population 10,000 Population t years after 2004 = Population P(t) = 10,000 + t(500) 10,000 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08 20 Population Growth (Cont.) Population is 10,000 in 2004; increases by 500 per year P(t) = 10,000 + t(500) P is a linear function of t . What is the slope? 500 people/year What is the y -intercept? number of people at time 0 (the year 2004) = 10,000 When P increases by a constant When number of people per year, P is a number linear function of t. linear Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 21 7 Population Growth (Cont.) Suppose a population is 10,000 in Jan 2004 and increases by 5% per year. Jan 2005? 10,000 + .05(10,000) = 10,000 + 500 = 10,500 Jan 2006? 10,500 + .05(10,500) = 10,500 + 525 = 11,025 Jan 2007? 11,025 + .05(11,025) = 11,025 + 551.25 = 11,576.25 Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 22 Population Growth (Cont.) Suppose a population is 10,000 in Jan 2004 and increases by Suppose i ncreases 5% per year. Let t be the number of years after 2004. Let P(t ) be the Let population in year t . What is the symbolic population What representation for P(t )? We know… representation Population in 2004 = P(0) = 10,000 Population Population in 2005 = P( 1 ) = 10,000 + .05 (10,000) = Population 1.05(10,000) = 1.051 (10,000) =10,500 1.05(10,000) 1 .05 Population in 2006 = P( 2 ) = 10,500 + .05 (10,500) = Population 1.05 (10,500) = 1.05 (1.05)(10,000) = 1.052(10,000) 1.05 (1.05)(10,000) 1.05 = 11,025 Population t years after 2004 = P(t) = 10,000(1.05)t Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 23 Population Growth (Cont.) Population is 10,000 in 2004; increases by 5% per year P(t) = 10,000 (1.05) t P is an E XPONENTIAL function of t. More specifically, an exponential growth function. What is the b ase of the exponential function? 1.05 What is the y-intercept? number of people at time 0 (the year 2004) = 10,000 When P increases by a constant percentage per year, P is an exponential function of t. Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 24 8 The Main Difference Between a Linear Growth and an Exponential Growth • A Linear Function adds a L inear adds fixed amount to the fixed previous value of y for previous each unit increase in x each • For example, in f ( x) For = 10,000 + 500x 500 is added to y for each added increase of 1 in x. • An Exponential Function An Exponential multiplies a fixed amount to the previous value of y to for each unit increase in x. • For example, in f(x ) = 10,000 (1.05)x y is multiplied by 1.05 for each increase of 1 in x. each http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08 25 The Definition of an Exponential Function A function represented by f(x ) = Cax, a > 0, a is not 1, and C > 0 is an Ca exponential function with base a and coefficient C . If a > 1, then f is an exponential growth function If growth If 0 < a < 1, then f is an exponential decay function If decay http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08 26 What is the Common Mistake? Don’t confuse f (x) = 2x with f (x ) = x2 f(x ) = 2x is an exponential function. f(x ) = x2 is a polynomial function, specifically a quadratic function. The functions and consequently their graphs are very different. f(x ) = x2 f(x ) = 2x Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 27 9 Exponential Growth vs. Decay • Example of exponential • Example of exponential decay function growth function f (x) = 3 • 2x f(x) Recall, in the exponential function f ( x) = C a x Recall, If a > 1, then f is an exponential growth function If If 0 < a < 1, then f is an exponential decay function If http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08 28 Properties of an Exponential Growth Function Example f ( x ) = 3 • 2x Rev.S08 Properties of an exponential growth function: •Domain: (-∞ , ∞) •Range: (0, ∞) Range: •f increases on (-∞, ∞) •The negative x-axis is a The horizontal asymptote. •y-intercept is (0,3). http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 29 Properties of an Exponential Decay Function Example Rev.S08 Properties of an exponential decay function: • Domain: (-∞ , ∞) • Range: (0, ∞) Range: • f decreases on (-∞, ∞) decreases • The positive x-axis is a The horizontal asymptote. • y-intercept is (0,3). http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 30 10 Example of an Exponential Decay: Carbon-14 Dating The time it takes for half of the atoms to decay into a h alf different element is called the half-life of an element different half-life undergoing radioactive decay. The half-life of carbon-14 is 5700 years. The h alf-life Suppose C grams of carbon-14 are present at t = 0. Suppose Then after 5700 years there will be C /2 grams present. Then Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 31 Example of an Exponential Decay: Carbon-14 Dating (Cont.) Let t be the number of years. Let A =f ( t ) be the amount of carbon-14 present at time t . Let C be the amount of carbon-14 present at t = 0. Then f(0) = C and f (5700) = C/2. Thus two points of f are (0,C) and (5700, C /2) Using the point (5700, C/2) and substituting 5700 for t and C/2 for A in A = f (t ) = Ca t yields: C/2 = C a 5700 Dividing both sides by C yields: 1/2 = a 5700 Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 32 Example of an Exponential Decay: Carbon-14 Dating (Cont.) Half-life Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 33 11 Radioactive Decay (An Exponential Decay Model) If a radioactive sample containing C units has a half-life h alf-life of k years, then the amount A remaining after x years of is given by Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 34 Example of Radioactive Decay Radioactive strontium-90 has a half-life of about 28 years and sometimes contaminates the soil. After 50 years, what percentage of a sample of radioactive strontium would remain? Note calculator keystrokes: Since C is present initially and after 50 years .29C remains, then 29% remains. Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 35 Example of an Exponential Growth: Compound Interest Suppose $10,000 is deposited into an account which pays 5% interest compounded annually. Then the amount A in 5% c ompounded Then the account after t years is: the A (t) = 10,000 (1.05)t Note the similarity with: Suppose a population is 10,000 in 2004 and increases by 5% per year. Then the population P, t years after 2004 is: P (t) = 10,000 (1.05)t Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 36 12 What is the Natural Exponential Function? The function f , represented by The f (x ) = e x is the natural exponential function where n atural e ≈ 2.718281828 Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 37 Example of Using Natural Exponential Function Suppose $100 is invested in an account with an interest rate of 8% c ompounded continuously. How much money will there be in the account after 15 years? In this case, P = $100, r = 8/100 = 0.08 and t = 15 years. Thus, A = P ert A = $ 100 e.08(15) A = $ 332.01 Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 38 Logarithmic Functions and Models Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 39 13 What is the Definition of a Common Logarithmic Function? The common logarithm of a positive number x, denoted log (x), is defined by log (x) = k if and only if x = 10k where k is a real number. The function given by f (x ) = log (x) is called the common logarithmic function. Note that the input x must be positive. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08 40 Let’s Evaluate Some Common Logarithms log (10) 1 because 101 = 10 log (100) 2 because 102 = 100 log (1000) 3 because 103 = 1000 log (10000) 4 because 104 = 10000 log (1/10) –1 because 10-1 = 1/10 log (1/100) –2 because 10-2 = 1/100 log (1/1000) –3 because 10-3 = 1/1000 log (1) 0 because 100 = 1 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08 41 Graph of a Logarithmic Function x f(x ) 0.01 -2 0.1 -1 1 0 10 1 100 2 Rev.S08 Note that the graph of y = log (x) is the Note graph of y = 10x reflected through the reflected lline y = x . This suggests that these are ine inverse functions. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 42 14 What is the Inverse Function of a Common Logarithmic Function? Note that the graph of f (x) = log (x) passes the horizontal line test so it is a one-to-one function and has an inverse function. Find the inverse of y = log (x) Using the definition of common logarithm to solve for x gives x = 10 y Interchanging x and y gives y = 10 x Thus, the inverse of y = log (x) is y = 10x Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 43 What is the Inverse Properties of the Common Logarithmic Function? Recall that f -1(x) = 10x given f(x ) = log (x) -1 Since (f ° f -1 )(x ) = x for every x in the domain of f log(10x) = x for all real numbers x . -1 -1 -1 Since (f -1 ° f)(x) = x for every x in the domain of f 10logx = x for any positive number x Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 44 What is the Definition of a Logarithmic Function with base a? • The logarithm with base a of a positive number x, The denoted by loga(x) is defined by denoted log is loga(x) = k if and only if x = a k w here a > 0, a ≠ 1, and k is a real number. where and • The function given by f (x ) = loga (x) is called the The llogarithmic function with base a . ogarithmic Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 45 15 What is the Natural Logarithmic Function? • Logarithmic Functions with Base 10 are called “common logs.” logs • log (x) means log10(x) - T he Common Logarithmic Function The • Logarithmic Functions with Base e are called “natural Logarithmic logs. ” • ln (x) means loge (x) - The Natural Logarithmic Function The http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08 46 Let’s Evaluate Some Natural Logarithms • ln (e) ln (e) = loge(e) = 1 since e 1 = e • ln (1) ln(e 2 ) = loge (e 2) = 2 since 2 is the exponent that goes on e to produce e2. ln (1) = loge1 = 0 since e0= 1 •. 1/2 since 1/2 is the exponent that goes on e to produce e 1/2 • ln (e 2) Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 47 What is the Inverse of a Logarithmic Function with base a? Note that the graph of f (x) = loga (x) passes the horizontal line test so it is a one-to-one function and has an inverse function. Find the inverse of y = loga(x) Using the definition of common logarithm to solve for x gives x = ay Interchanging x and y gives y = ax Thus, the inverse of y = loga (x) is y = ax Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 48 16 What is the Inverse Properties of a Logarithmic Function with base a? Recall that f -1(x) = a x given f ( x) = loga (x) -1 Since (f ° f -1 )(x ) = x for every x in the domain of f loga ( a x ) = x for all real numbers x . -1 -1 -1 Since (f -1 ° f)(x) = x for every x in the domain of f loga x = x for any positive number x a Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 49 Let’s Try to Solve Some Equations Solve the equation 4x = 1/64 the Take the log of both sides to the base 4 base log4 (4x) = log4 (1/64) (4 log Using the inverse property loga (ax) =x , t his simplifies to Using the log ( a x = log4(1/64) log Since 1/64 can be rewritten as 4 –3 Since 1/64 x = log4(4 –3) log Using the inverse property loga ( a x) = x , this simplifies to Using the log x = –3 Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 50 Let’s Try to Solve Some Equations Solve the equation ex = 15 Take the log of both sides to the base e base ln( e x ) = ln(15) ln Using the inverse property loga (ax) = x this simplifies to Using the log x = ln(15) ln(15) Using the calculator to estimate ln (15) x ≈ 2.71 Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 51 17 Let’s Try to Solve Some Equations (Cont.) Solve the equation ln (x) = 1.5 the Exponentiate both sides using base e elnx = e1.5 Using the inverse property a loga x = x t his simplifies to Using the x = e1.5 Using the calculator to estimate e 1.5 Using x ≈ 4.48 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08 52 What have we learned? We have learned to: 1. 2. 3. 4. 5. 6. 7. Distinguish between linear and exponential growth. Model data with exponential functions. Calculate compound interest. Use the natural exponential function in applications. Evaluate the common logarithmic function. Evaluate the natural logarithmic function. Solve basic exponential and logarithmic equations. Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 53 Credit Some of these slides have been adapted/modified in part/whole from the slides of the following textbook: • Rockswold, Gary, Precalculus with Modeling and Visualization, 3th Edition Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 54 18 ...
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This note was uploaded on 10/18/2011 for the course MAC 1105 taught by Professor Russel during the Summer '07 term at Valencia.

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