ca_m8 - MAC 1105 Module 8 Exponential and Logarithmic...

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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 A Quick Review on Function y = f(x) means that given an input x, there is means there just one corresponding output y. Graphically, this means that the graph passes the vertical line test. vertical Numerically, this means that in a table of Numerically, values for y = f(x) there are no x-values there -values repeated. 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 )? of x? 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 -values are repeated.) are 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 yvalues repeated in a table of values. Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 6 Example Given y = f(x) = |x|, is f one-to-one? |, 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 1 0 0 1 1 2 2 Note Note Rev.S08 (-2,2) (2,2) that there is a value of y in the table for which there are two different values of x (that is, yvalues 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 one-to-one and d in the domain of f, 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 not be confused with the reciprocal. be • 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) is pronounced “y equals f Note inverse of x.” Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 9 Example of an Inverse Function Let F be Fahrenheit temperature and let C be Centigrade temperature. 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 -1 Note that the domain of f equals the range of f -1 and the Note -1 range of f equals the domain of f -1 . Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 12 How to Find Inverse Function Symbolically? -1 Check that f is a one-to-one function. If not, f -1 does one-to-one If not exist. not Solve the equation y = f(x) for x, resulting in the Solve for resulting -1 -1(y) equation x = f -1 Interchange x and y to obtain y = f -1(x) Interchange Example. Example. f(x) = 3x + 2 y = 3x + 2 Solving for x gives: 3x = y – 2 Solving x = (y – 2)/3 2)/3 Interchanging x and y gives: y = (x – 2)/3 Interchanging 2)/3 So f -1(x) = (x – 2)/3 So -1 Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 13 How to Evaluate Inverse Function Numerically? x f(x) 1 –5 2 –3 3 0 4 3 5 5 Rev.S08 -1 The function is one-to-one, so f -1 one-to-one exists. 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. 14 How to Evaluate Inverse Function Graphically? The graph of f below The passes the horizontal line test so f is one-toline one-toone. -1 Evaluate f -1(4). Since the point (2,4) is on Since the graph of f, the point the (4,2) will be on the graph -1 -1 of f -1 and thus f -1(4) = 2. Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. f(2)=4 15 What is the Formal Definition of Inverse Functions? -1 Let f be a one-to-one function. Then f -1 is the inverse function of f, iif f -1 -1 • (f -1 o f)(x) = f -1(f(x)) = x for every x in the )) domain of f -1 -1 • (f o f -1 )(x) = f(f -1(x)) = x for every x in the )) -1 domain of f -1 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 a population is 10,000 in January 2004. Suppose the population increases by… 500 people per year What is the population in Jan What 2005? 10,000 + 500 = 10,500 10,500 What is the population in Jan What 2006? 2006? 10,500 + 500 = 11,000 10,500 11,000 Rev.S08 5% per year What is the population in What Jan 2005? Jan 10,000 + .05(10,000) = 10,000 10,000 + 500 = 10,500 10,500 What is the population in What Jan 2006? Jan 10,500 + .05(10,500) = 10,500 10,500 + 525 = 11,025 11,025 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 18 Population Growth (Cont.) Suppose a population is 10,000 in Jan 2004. Suppose the Suppose population increases by 500 per year. What is the by population in …. population Jan 2005? Jan 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 Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 19 Population Growth (Cont.) Suppose a population is 10,000 in Jan 2004 and increases by 500 per year. Let t be the number of years after 2004. Let P(t) be the population in year t. What is the symbolic representation for P(t)? We know… 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 Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 20 Population Growth (Cont.) Population is 10,000 in 2004; increases by 500 per year 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 linear function of t. Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 21 Population Growth (Cont.) Suppose a population is 10,000 in Jan 2004 and 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 = 11,025 11,576.25 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 Suppose a population is 10,000 in Jan 2004 and by 5% per year. Let t be the number of years after 2004. Let P(t) be the Let be population in year t. What is the symbolic representation for P(t)? We know… Population in 2004 = P(0) = 10,000 Population (0) Population in 2005 = P(1) = 10,000 + .05 (10,000) = 10,000 1.05(10,000) = 1.051(10,000) =10,500 1.05 Population in 2006 = P(2) = 10,500 + .05 (10,500) = Population 10,500 1.05 (10,500) = 1.05 (1.05)(10,000) = 1.052(10,000) 1.05 1.05 (1.05)(10,000) 1.05 = 11,025 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 Population is 10,000 in 2004; increases by 5% per year P(t) = 10,000 (1.05)t P is an EXPONENTIAL function of t. More specifically, an exponential growth function. What is the base 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 When percentage per year, P is an exponential function of t. exponential Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 24 The Main Difference Between a Linear Growth and an Exponential Growth • A Linear Function adds a Linear fixed amount to the previous value of y for each unit increase in x • For example, in f(x) For = 10,000 + 500x 500 is 10,000 added to y for each increase of 1 in x. increase Rev.S08 • An Exponential Function An Exponential multiplies a fixed amount multiplies to the previous value of y for each unit increase in x. • For example, in For f(x) = 10,000 (1.05)x y is multiplied by 1.05 for each increase of 1 in x. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 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. exponential 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 Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 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 Exponential Growth vs. Decay • Example of exponential • Example of exponential Example Example decay function decay growth function growth f(x) = 3 2x f(x) • Recall, in the exponential function f(x) = Cax Recall, If a > 1, then f is an exponential growth function If If 0 < a < 1, then f is an exponential decay function If Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 28 Properties of an Exponential Growth Function Example f(x) = 3 2x • Rev.S08 Properties of an exponential growth function: •Domain: (-∞, ∞) •Range: (0, ∞) •f increases on (-∞, ∞) •The negative x-axis is a The -axis horizontal asymptote. horizontal •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, ∞) • f decreases on (-∞, ∞) decreases • The positive x-axis is a The -axis horizontal asymptote. horizontal • y-intercept is (0,3). http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 30 Example of an Exponential Decay: Carbon-14 Dating The time it takes for half of the atoms to decay into a The half different element is called the half-life of an element half-life undergoing radioactive decay. The half-life of carbon-14 is 5700 years. half-life Suppose C grams of carbon-14 are present at t = 0. Suppose Then after 5700 years there will be C/2 grams present. 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) = Cat yields: C/2 = C a5700 Dividing both sideshttp://faculty.valenciacc.edu/ashaw/ a5700 by C yields: 1/2 = Rev.S08 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 Radioactive Decay (An Exponential Decay Model) If a radioactive sample containing C units has a half-life half-life of k years, then the amount A remaining after x years is given by is 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? would Note calculator Note keystrokes: keystrokes Since C is present initially and after 50 years .29C remains, Since then 29% remains. then 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 compounded Then the account after t years is: A(t) = 10,000 (1.05)t Note the similarity with: Suppose a population is 10,000 in Note 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 What is the Natural Exponential Function? The function f, represented by represented f(x) = ex is the natural exponential function where natural 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% compounded 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 = Pert 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 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. Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 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 Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 41 Graph of a Logarithmic Function x f(x) 0.0 -2 1 0.1 1 10 100 Rev.S08 -1 0 1 2 Note that the graph of y = log (x) is the graph of y = 10x reflected through the line graph y = x. This suggests that these are This inverse functions. inverse http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 42 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 = 10y Interchanging x and y gives y = 10x 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 -1 Since (f ° f -1 )(x) = x for every x in the domain of f -1 log(10x) = x for all real numbers x. log( -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 log is loga(x) = k if and only if x = ak where a > 0, a ≠1, and k is a real number. where The • The function given by f(x) = loga(x) is called the logarithmic function with base a. Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 45 What is the Natural Logarithmic Function? • Logarithmic Functions with Base 10 are called “common logs.” logs.” The • log (x) means log10(x) - The Common Logarithmic Function • Logarithmic Functions with Base e are called “natural Logarithmic natural logs.” logs.” The • ln (x) means loge(x) - The Natural Logarithmic Function Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 46 Let’s Evaluate Some Natural Logarithms • ln (e) ln (e) = loge(e) = 1 since e1= e • ln (1) ln(e2) = loge (e2) = 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 e1/2 • ln (e2) 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 What is the Inverse Properties of a Logarithmic Function with base a? Recall that f -1(x) = ax given f(x) = loga(x) -1 -1 Since (f ° f -1 )(x) = x for every x in the domain of f -1 loga(ax) = x for all real numbers x. log -1 Since (f -1 ° f)(x) = x for every x in the domain of f alogax = x for any positive number x 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 Solve 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 , this 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 (ax) = 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 Solve Take the log of both sides to the base e base ln(ex) = 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 Let’s Try to Solve Some Equations (Cont.) Solve the equation ln (x) = 1.5 Solve the Exponentiate both sides using base e elnx = e1.5 Using the inverse property alogax = x this simplifies to Using the x = e1.5 Using the calculator to estimate e1.5 Using x ≈ 4.48 Rev.S08 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 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 ...
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This note was uploaded on 11/11/2011 for the course MATH 110 taught by Professor Staff during the Winter '08 term at BYU.

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