Exponential and Logarithmic Functions

Transcription

1 Eponential and Logarithmic Functions 0 0. Algebra and Composition of Functions 0. Inverse Functions 0. Eponential Functions 0. Logarithmic Functions 0. Properties of Logarithms 0. The Irrational Number e Problem Recognition Eercises Logarithmic and Eponential Forms 0.7 Logarithmic and Eponential Equations Chapter 0 is devoted to the stud eponential and logarithmic functions. These functions are used to stud man naturall occurring phenomena such as population growth, eponential deca of radioactive matter, and growth of investments. The following is a Sudoku puzzle. As ou work through this chapter, tr to simplif the epressions or solve the equations in the clues given below. Use the clues to fill in the boes labeled a n. Then fill in the remaining part of the grid so that ever row, ever column, and ever bo contains the digits through. Clues a. log b. In e c. Solution to 0 9 d. Solution to ln ln e. log f. log log g. Solution to log log h. ln ln i. ff j. ln e k. log 7 7 e f i l a m c d g j b h k l. m. a e 0 b n n. log 00 9

2 90 Chapter 0 Eponential and Logarithmic Functions Section 0. Concepts. Algebra of Functions. Composition of Functions. Multiple Operations on Functions Algebra and Composition of Functions. Algebra of Functions Addition, subtraction, multiplication, and division can be used to create a new function from two or more functions. The domain of the new function will be the intersection of the domains of the original functions. Finding domains of functions was first introduced in Section.. Sum, Difference, Product, and Quotient of Functions f Given two functions f and g, the functions f g, f g, f g, and g are defined as f g f g f g f g f g f g a f f b g g provided g 0 ( f g)() 7 f () g() Figure 0- f () g() For eample, suppose f ƒ ƒ and g. Taking the sum of the functions produces a new function denoted b f g. In this case, f g ƒ ƒ. Graphicall, the -values of the function f g are given b the sum of the corresponding -values of f and g. This is depicted in Figure 0-. The function f g appears in red. In particular, notice that f g f g. Eample Adding, Subtracting, and Multipling Functions Given: g h k a. Find g h and write the domain of g h in interval notation. b. Find h g and write the domain of h g in interval notation. c. Find g k and write the domain of g k in interval notation. Solution: a. b. c. g h g h h g h g 7 g k g k The domain is all real numbers,. The domain is all real numbers,. The domain is, because 0 for.

3 Section 0. Algebra and Composition of Functions 9 Skill Practice Given: f g h Perform the indicated operations. Write the domain of the resulting function in interval notation.. f g. g f. f h Eample Dividing Functions Given the functions defined b h and k, find Q k h R and write the domain of Q k h R in interval notation. Solution: a k b h To find the domain, we must consider the restrictions on imposed b the square root and b the fraction. From the numerator we have 0 or, equivalentl,. From the denominator we have 0 or, equivalentl, 0. Hence, and 0. k h Thus, the domain of is the set of real numbers greater than or equal to, but not equal to or 0. This is shown graphicall in Figure 0-. The domain is [, ) (, ). [ 0 Figure 0- Skill Practice Given: f g. Find a f and write the domain interval notation. g b. Composition of Functions Composition of Functions The composition of f and g, denoted f g, is defined b the rule f g f g provided that g is in the domain of f The composition of g and f, denoted g f, is defined b the rule g f g f provided that f is in the domain of g Note: f g is also read as f compose g, and g f is also read as g compose f. Skill Practice Answers. ; domain:,. ; domain:,. ; domain:,. ; domain:,0 0,

4 9 Chapter 0 Eponential and Logarithmic Functions For eample, given f and g, we have f g f g f 0 7 Substitute g into the function f. In this composition, the function g is the innermost operation and acts on first. Then the output value of function g becomes the domain element of the function f, as shown in the figure. g f g() f(g()) Eample Composing Functions Given: f, g, and n a. Find f g and write the domain of f g in interval notation. b. Find g f and write the domain of g f in interval notation. c. Find n f and write the domain of n f in interval notation. TIP: Eamples (a) and (b) illustrate that the order in which two functions are composed ma result in different functions. That is, f g does not necessaril equal g f. Skill Practice Answers. ; domain:,. ; domain:, 7. ; domain:, Solution: a. b. c. f g f g f Evaluate the function f at. Replace b in the function f. The domain is all real numbers,. g f g f g Evaluate the function g at. Replace b in function g. 0 The domain is all real numbers,. n f n f n Evaluate the function n at. Replace b the quantit in function n. The domain is,. Skill Practice Given f, g, and h,. Find f g. Write the domain of f g in interval notation.. Find g f. Write the domain of g f in interval notation. 7. Find h g. Write the domain of h g in interval notation.

5 Section 0. Algebra and Composition of Functions 9. Multiple Operations on Functions Eample Combining Functions Given the functions defined b values, if possible. f 7 and h, find the function a. f h b. a h c. f b 7 h f Solution: a. f h f h f h() is a product (not a composition). 7 7 h b. The function has restrictions on its domain. f a h h b f f 7 Therefore, is not in the domain, and a h h7 7 b 7 is undefined. f f 7 Avoiding Mistakes: h If ou had tried evaluating the function f at 7, the denominator would be zero and the function undefined. h7 7 f c. h f h f Evaluate f() first. f 7. h Substitute the result into function h. 0 Skill Practice Given: h k. Find h k. 9. Find a h 0. Find k h. k b. Skill Practice Answers

6 9 Chapter 0 Eponential and Logarithmic Functions Eample Finding Function Values from a Graph For the functions f and g pictured, find the function values if possible. a. g() b. c. d. f g a g f b f g Solution: a. g The value g() represents the -value of g (the red graph) when. Because the point (, ) lies on the graph, g. b. f g f g Evaluate the difference of f and g. Estimate function values from the graph. g() f() c. a g g b f f Evaluate the quotient of g() and f(). (undefined) 0 g f The function is undefined at because the denominator is zero. d. f g f g f 0 From the red graph, find the value of g first. From the blue graph, find the value of f at 0. Skill Practice Find the values from the graph. f() g() Skill Practice Answers g. f g. a f. g f g b

7 Section 0. Algebra and Composition of Functions 9 Section 0. Boost our GRADE at mathzone.com! Stud Skills Eercise Practice Eercises. Define the ke term composition of functions. Practice Problems Self-Tests NetTutor e-professors Videos Concept : Algebra of Functions For Eercises, refer to the functions defined below. f g h k Find the indicated functions. Write the domain in interval notation.. f g. f g.. f h. f h 7. g f h k. g f 9. f k 0. a h f b. ag. a f. f b g b a f h b Concept : Composition of Functions For Eercises, find the indicated functions and their domains. Use f, g, h, and k as defined in Eercises.. f g. f k. g f 7. k f. k h k g. g k. h k f h. Based on our answers to Eercises and 7 is it true in general that f k k f?. Based on our answers to Eercises and is it true in general that f g g f? For Eercises 0, find f g and g f.. f, g. f, g 7. f, g. f g 9. For h, 0. For k,, find h h. find k k.

8 9 Chapter 0 Eponential and Logarithmic Functions Concept : Multiple Operations on Functions For Eercises, refer to the functions defined below. m r n p Find the function values if possible.. m r0. n p0. m r.. r n. n r 7. p m. 9. m p 0. r m. r p. n m m n n p. m p. r m For Eercises, approimate the function values from the graph, if possible.. f. f() 7. g. g() 9. f g 0. g f. f g. g f. a g f b 0. a f. a f. a g g b g b 0 f b g() f() 7. g f. f g g f. g g. f g f f For Eercises 7, approimate the function values from the graph, if possible.. a. a(). b. b() 7. a b. a b0 9. b a 70. a b 7. b a0 7. a b 7. a b 7. b a a() b() 7. ab 7. a a 77. a a 7. a b b b b b 79. The cost in dollars of producing to cars is C.. The revenue received is R.9. To calculate profit, subtract the cost from the revenue. a. Write and simplif a function P that represents profit in terms of. b. Find the profit of producing 0 to cars.

9 Section 0. Algebra and Composition of Functions The cost in dollars of producing lawn chairs is C. 0.. The revenue for selling chairs is R.99. To calculate profit, subtract the cost from the revenue. a. Write and simplif a function P that represents profit in terms of. b. Find the profit in producing 00 lawn chairs.. The functions defined b Dt 0.9t.9 and Rt 0.7t 0. approimate the amount of child support (in billions of dollars) that was due and the amount of child support actuall received in the United States between the ears 000 and 00. In each case, t 0 corresponds to the ear 000. a. Find the function F defined b Ft Dt Rt. What does F represent in the contet of this problem? b. Find F(0), F(), and F(). What do these function values represent in the contet of this problem? Amount ($ billions) Difference Between Child Support Due and Child Support Paid, United States, F(t) 0.t. 0 0 Year (t 0 corresponds to 000) D(t) 0.9t.9 (Source: U.S. Bureau of the Census) R(t) 0.7t 0.. If t represents the number of ears after 900, then the rural and urban populations in the South (United States) between the ears 900 and 970 can be approimated b rt.97t.t 0,0 where t 0 corresponds to the ear 900 and r(t) represents the rural population in thousands. ut 0.0t 0.9t 77.t 9 where t 0 corresponds to the ear 900 and u(t) (Source: Historical Abstract of the United States) represents the urban population in thousands. a. Find the function T defined b T(t) r(t) u(t). What does the function T represent in the contet of this problem? b. Use the function T to approimate the total population in the South for the ear 90.. Joe rides a biccle and his wheels revolve at 0 revolutions per minute (rpm). Therefore, the total number of revolutions, r, is given b rt 0t, where t is the time in minutes. For each revolution of the wheels of the bike, he travels approimatel 7 ft. Therefore, the total distance he travels, D, depends on the total number of revolutions, r, according to the function Dr 7r. a. Find D r t and interpret its meaning in the contet of this problem. b. Find Joe s total distance in feet after 0 min.. The area, A, of a square is given b the function a, where is the length of the sides of the square. If carpeting costs $9.9 per square ard, then the cost, C, to carpet a square room is given b Ca 9.9a, where a is the area of the room in square ards. Population (thousands) a. Find C a and interpret its meaning in the contet of this problem. Rural and Urban Populations in the South, United States, b. Find the cost to carpet a square room if its floor dimensions are d b d.,000 0,000,000 0,000 r(t).97t.t 0,0,000 0,000,000 0, u(t) 0.0t 0.9t 77.t 9 0 t Year (t 0 corresponds to 900)

10 9 Chapter 0 Eponential and Logarithmic Functions Section 0. Concepts. Introduction to Inverse Functions. Definition of a One-to-One Function. Finding an Equation of the Inverse of a Function. Definition of the Inverse of a Function Avoiding Mistakes: f denotes the inverse of a function. The does not represent an eponent. Inverse Functions. Introduction to Inverse Functions In Section., we defined a function as a set of ordered pairs (, ) such that for ever element in the domain, there corresponds eactl one element in the range. For eample, the function f relates the price, (in dollars), of a USB Flash drive to the amount of memor that it holds, (in megabtes). price, $ megabtes, f {(, ), (9, ), (, )} That is, the amount of memor depends on how much mone a person has to spend. Now suppose we create a new function in which the values of and are interchanged. The new function is called the inverse of f and is denoted b f. This relates the amount of memor,, to the cost,. price, megabtes, $ f {(, ), (, 9), (, )} Notice that interchanging the - and -values has the following outcome. The domain of f is the same as the range of f, and the range of f is the domain of f.. Definition of a One-to-One Function A necessar condition for a function f to have an inverse function is that no two ordered pairs in f have different -coordinates and the same -coordinate. A function that satisfies this condition is called a one-to-one function. The function relating the price of a USB Flash drive to its memor is a one-to-one function. However, consider the function g defined b g {(, ), (, ), (, )} same different This function is not one-to-one because the range element has two different -coordinates, and. Interchanging the - and -values produces a relation that violates the definition of a function. {(, ), (, ), (, )} same different This relation is not a function because for there are two different -values, and. In Section., ou learned the vertical line test to determine visuall if a graph represents a function. We use a horizontal line test to determine whether a function is one-to-one.

11 Section 0. Inverse Functions 99 Horizontal Line Test Consider a function defined b a set of points (, ) in a rectangular coordinate sstem. The graph of the ordered pairs defines as a one-to-one function of if no horizontal line intersects the graph in more than one point. To understand the horizontal line test, consider the functions f and g. f {(,.99), (.,.9), (,.9)} g {(, ), (, ), (, )} This function is one-to-one. No horizontal line intersects more than once. This function is not one-to-one. A horizontal line intersects more than once. Eample Identifing One-to-One Functions Determine whether the function is one-to-one. a. b. Solution: a. Function is not one-to-one. b. Function is one-to-one. A horizontal line intersects No horizontal line intersects more in more than one point. than once.

12 700 Chapter 0 Eponential and Logarithmic Functions Skill Practice Use the horizontal line test to determine if the functions are oneto-one.... Finding an Equation of the Inverse of a Function Another wa to view the construction of the inverse of a function is to find a function that performs the inverse operations in the reverse order. For eample, the function defined b f() multiplies b and then adds. Therefore, the inverse function must subtract from and divide b. We have f To facilitate the process of finding an equation of the inverse of a one-to-one function, we offer the following steps. Finding an Equation of an Inverse of a Function For a one-to-one function defined b f(), the equation of the inverse can be found as follows:. Replace f() b.. Interchange and.. Solve for.. Replace b f (). Eample Find the inverse. f() Finding an Equation of the Inverse of a Function Solution: We know the graph of f is a nonvertical line. Therefore, f() defines a one-to-one function. To find the inverse we have Step : Replace f() b. Step : Interchange and. Skill Practice Answers. Not one-to-one. One-to-one

13 Section 0. Inverse Functions 70 f Step : Solve for. Subtract from both sides. Divide both sides b. Step : Replace b f (). Skill Practice. Find the inverse of f (). The ke step in determining the equation of the inverse of a function is to interchange and. B so doing, a point (a, b) on f corresponds to a point (b, a) on f. For this reason, the graphs of f and f are smmetric with respect to the line (Figure 0-). Notice that the point (, ) of the function f corresponds to the point (, ) of f. Likewise, (, ) of f corresponds to (, ) of f. f() (, ) f () (, ) (, ) (, ) Figure 0- Eample Finding an Equation of the Inverse of a Function Find the inverse of the one-to-one function. g Solution: g Step : Replace g() b. Step : Interchange and. Step : Solve for. Add to both sides. To eliminate the cube root, cube both sides. Simplif the right side. Divide both sides b. Step : Replace b g (). Skill Practice. Find the inverse of h. Skill Practice Answers. f. h

14 70 Chapter 0 Eponential and Logarithmic Functions The graphs of g and g from Eample are shown in Figure 0-. Once again we see that the graphs of a function and its inverse are smmetric with respect to the line. For a function that is not one-to-one, sometimes we can restrict its domain to create a new function that is one-to-one. This is demonstrated in Eample. Eample g ( ) () Figure 0- Finding the Equation of an Inverse of a Function with a Restricted Domain Given the function defined b m() for 0, find an equation defining m. Solution: From Section., we know that is a parabola with verte at (0, ) (Figure 0-). The graph represents a function that is not one-to-one. However, with the restriction on the domain 0, the graph of m(), 0, consists of onl the right branch of the parabola (Figure 0-). This is a one-to-one function. g() m() ; 0 0 Figure 0- Figure 0- To find the inverse, we have Step : Replace m() b. Step : Interchange and. Notice that the restriction 0 becomes 0. Step : Solve for. Subtract from both sides. Appl the square root propert. Notice that we obtain the positive square root of because of the restriction 0.

15 Section 0. Inverse Functions 70 m Figure 0-7 shows the graphs of m and m. Compare the domain and range of m and m. Domain of m: 0, Range of m:, Domain of m :, Range of m : 0, Skill Practice. Find the inverse. g() 0 Step : Replace b m (). Notice that the domain of m has the same values as the range of m m() ; 0 m () 0 Figure 0-7. Definition of the Inverse of a Function Definition of an Inverse Function If f is a one-to-one function represented b ordered pairs of the form (, ), then the inverse function, denoted f, is the set of ordered pairs denoted b ordered pairs of the form (, ). An important relationship between a function and its inverse is shown in Figure 0-. Domain of f Range of f f f Range of f Domain of f Figure 0- Recall that the domain of f is the range of f and the range of f is the domain of f. The operations performed b f are reversed b f. This leads to the inverse function propert. Skill Practice Answers. g

16 70 Chapter 0 Eponential and Logarithmic Functions Inverse Function Propert If f is a one-to-one function, then g is the inverse of f if and onl if (f g)() for all in the domain of g and (g f )() for all in the domain of f Eample Composing a Function with Its Inverse Show that the functions are inverses. Solution: h and k To show that the functions h and k are inverses, we need to confirm that (h k)() and k h. h k hk h a b a b h k as desired. k h kh k k h as desired. The functions h and k are inverses because (h k)() and (k h)() for all real numbers. Skill Practice Answers. f g f g a b g f g f Skill Practice. Show that the functions are inverses. f and g

17 Section 0. Inverse Functions 70 Section 0. Boost our GRADE at mathzone.com! Stud Skills Eercise. Define the ke terms. Practice Eercises Practice Problems Self-Tests NetTutor e-professors Videos a. Inverse function b. One-to-one function c. Horizontal line test Review Eercises. Write the domain and range of the relation {(, ), (, ), (, ), (, 0)}. For Eercises, determine if the relation is a function b using the vertical line test. (See Section..) Concept : Introduction to Inverse Functions For Eercises 9, write the inverse function for each function. 9. g {(, ), (, ), (, 9), (0, )} 0. f {(, ), ( 9, 0), (, ), (, )}. r {(a, ), (b, ), (c, 9)}. s {(, ), (, ), (, z)} Concept : Definition of a One-to-One Function. The table relates a state,, to the number of representatives in the House of Representatives,, in the ear 00. Does this relation define a one-to-one function? If so, write a function defining the inverse as a set of ordered pairs. Number of State Representatives Colorado 7 California Teas Louisiana 7 Pennslvania 9

18 70 Chapter 0 Eponential and Logarithmic Functions. The table relates a cit to its average Januar temperature. Does this relation define a one-to-one function? If so, write a function defining the inverse as a set of ordered pairs. Cit Temperature ( C) Gainesville, Florida. Keene, New Hampshire.0 Wooster, Ohio.0 Rock Springs, Woming.0 Lafaette, Louisiana 0.9 For Eercises 0, determine if the function is one-to-one b using the horizontal line test Concept : Finding an Equation of the Inverse of a Function For Eercises 0, write an equation of the inverse for each one-to-one function as defined.. h(). k(). m. n(). p() 0. q 7. f(). g 9. g 0. f. The function defined b f() 0.0 converts a length of feet into f() meters. a. Find the equivalent length in meters for a -ft board and a 0-ft wire. b. Find an equation defining f (). c. Use the inverse function from part (b) to find the equivalent length in feet for a 00-m race in track and field. Round to the nearest tenth of a foot.

19 Section 0. Inverse Functions 707. The function defined b s().7 converts a speed of mph to s() ft/sec. a. Find the equivalent speed in feet per second for a car traveling 0 mph. b. Find an equation defining s (). c. Use the inverse function from part (b) to find the equivalent speed in miles per hour for a train traveling ft/sec. Round to the nearest tenth. For Eercises 9, answer true or false.. The function defined b has an inverse function defined b.. The domain of an one-to-one function is the same as the domain of its inverse.. All linear functions with a nonzero slope have an inverse function.. The function defined b g 0 0 is one-to-one. 7. The function defined b k() is one-to-one.. The function defined b h 0 0 for 0 is one-to-one. 9. The function defined b L() for 0 is one-to-one. 0. Eplain how the domain and range of a one-to-one function and its inverse are related.. If (0, b) is the -intercept of a one-to-one function, what is the -intercept of its inverse?. If (a, 0) is the -intercept of a one-to-one function, what is the -intercept of its inverse?. Can ou think of an function that is its own inverse?. a. Find the domain and range of the function defined b f. b. Find the domain and range of the function defined b f (), 0.. a. Find the domain and range of the function defined b g(), 0. b. Find the domain and range of the function defined b g. For Eercises 9, the graph of f() is given. a. State the domain of f. b. State the range of f. c. State the domain of f. d. State the range of f. e. Graph the function defined b f (). The line is provided for our reference.. 7. f() f()

20 70 Chapter 0 Eponential and Logarithmic Functions. 9. f() f() Concept : Definition of the Inverse of a Function For Eercises 0, verif that f and g are inverse functions b showing that a. f g b. g f 0. f and g. f and g. and. f f g and g 7. f, 0, and g,. f, 0, and g, Epanding Your Skills For Eercises 7, write an equation of the inverse of the one-to-one function.. f 7. p. t 9. w 0. g n 9 0. g 0.. v. z 7. m 0 q u Graphing Calculator Eercises For Eercises 7, use a graphing calculator to graph each function on the standard viewing window defined b 0 0 and 0 0. Use the graph of the function to determine if the function is one-to-one on the interval 0 0. If the function is one-to-one, find its inverse and graph both functions on the standard viewing window.. f 9. k() 70. g() m()

21 Section 0. Eponential Functions 709 Eponential Functions. Definition of an Eponential Function The concept of a function was first introduced in Section.. Since then we have learned to recognize several categories of functions, including constant, linear, rational, and quadratic functions. In this section and the net, we will define two new tpes of functions called eponential and logarithmic functions. To introduce the concept of an eponential function, consider the following salar plans for a new job. Plan A pas $ million for a month s work. Plan B starts with on Da, and ever da thereafter the salar is doubled. At first glance, the million-dollar plan appears to be more favorable. Look, however, at Table 0-, which shows the dail paments for 0 das under plan B. Section 0. Concepts. Definition of an Eponential Function. Approimating Eponential Epressions with a Calculator. Graphs of Eponential Functions. Applications of Eponential Functions Table 0- Da Pament Da Pament Da Pament $0. $0,97. $0.9 $,9.0 $.9 $,.0 $. $7,77. $7. $,. $. $7,0. 7 $. 7 $0.7 7 $,,77. $. $. $,,. 9 $. 9 $. 9 $,, $0. 0 $0,.7 0 $0,77,. Notice that the salar on the 0th da for plan B is over $0 million. Taking the sum of the paments, we see the total salar for the 0-da period is $,7,.. The dail salar for plan B can be represented b the function, where is the number of das on the job and is the salar for that da. An interesting feature of this function is that for ever positive -unit change in, the -value doubles. The function is called an eponential function. Definition of an Eponential Function Let b be an real number such that b 7 0 and b Z. Then for an real number, a function of the form b is called an eponential function. An eponential function is easil recognized as a function with a constant base and variable eponent. Notice that the base of an eponential function must be a positive real number not equal to.

22 70 Chapter 0 Eponential and Logarithmic Functions Calculator Connections On a graphing calculator, use the ^ ke to approimate an epression with an irrational eponent. Calculator Connections. Approimating Eponential Epressions with a Calculator Up to this point, we have evaluated eponential epressions with integer eponents and with rational eponents, for eample, and. However, how do we evaluate an eponential epression with an irrational eponent such as p? In such a case, the eponent is a nonterminating and nonrepeating decimal. The value of p can be thought of as the limiting value of a sequence of approimations using rational eponents: An eponential epression can be evaluated at all rational numbers and at all irrational numbers. Hence, the domain of an eponential function is all real numbers. Eample Approimating Eponential Epressions with a Calculator Approimate the epressions. Round the answers to four decimal places. 7 a. b. c p Solution: a..0 b c Skill Practice 9 p Approimate the value of the epressions. Round the answers to four decimal places Graphs of Eponential Functions The functions defined b f, g, h, and k are all eamples of eponential functions. Eample illustrates the two general shapes of eponential functions. Eample Graphing an Eponential Function Graph the functions f and g. a. f b. g Skill Practice Answers

23 Section 0. Eponential Functions 7 Solution: Table 0- shows several function values f() and g() for both positive and negative values of. The graph is shown in Figure 0-9. Table 0- f() g() () 0 g() ( ) f() 0 Figure 0-9 Skill Practice Graph the functions... f a f b The graphs in Figure 0-9 illustrate several important features of eponential functions. Graphs of f () b The graph of an eponential function defined b f b b 7 0 and b Z has the following properties.. If b 7, f is an increasing eponential function, sometimes called an eponential growth function. If 0 b, f is a decreasing eponential function, sometimes called an eponential deca function.. The domain is the set of all real numbers,,.. The range is 0,.. The -ais is a horizontal asmptote.. The function passes through the point (0, ) because f 0 b 0. These properties indicate that the graph of an eponential function is an increasing function if the base is greater than. Furthermore, the base affects its steepness. Consider the graphs of f, h, and k (Figure 0-0). For ever positive -unit change in, f increases b times, h increases b times, and k increases b times (Table 0-). Skill Practice Answers.. f() f() ()

24 7 Chapter 0 Eponential and Logarithmic Functions Table 0- f() h() k() Figure 0-0 k() h() f() The graph of an eponential function is a decreasing function if the base is between 0 and. Consider the graphs of g, m, and n (Table 0- and Figure 0-). Table 0-0 g() () m() () n() () n() ( ) m() ( ) g() ( ) Figure 0- Marie and Pierre Curie. Applications of Eponential Functions Eponential growth and deca can be found in a variet of real-world phenomena; for eample, Population growth can often be modeled b an eponential function. The growth of an investment under compound interest increases eponentiall. The mass of a radioactive substance decreases eponentiall with time. The temperature of a cup of coffee decreases eponentiall as it approaches room temperature. A substance that undergoes radioactive deca is said to be radioactive. The half-life of a radioactive substance is the amount of time it takes for one-half of the original amount of the substance to change into something else. That is, after each half-life the amount of the original substance decreases b one-half. In 9, Marie Curie discovered the highl radioactive element radium. She shared the 90 Nobel Prize in phsics for her research on radioactivit and was awarded the 9 Nobel Prize in chemistr for her discover of radium and polonium. Radium (an isotope of radium) has a half-life of 0 ears and decas into radon (a radioactive gas).

25 Section 0. Eponential Functions 7 Eample Appling an Eponential Deca Function In a sample originall having g of radium, the amount of radium present after t ears is given b where A is the amount of radium in grams and t is the time in ears. a. How much radium will be present after 0 ears? b. How much radium will be present after 0 ears? c. How much radium will be present after 0 ears? Solution: At a b t 0 At t 0 a. A0 a 0 0 Substitute t 0. b After 0 ears ( half-life), 0. g remains. b. A0 a 0 0 Substitute t 0. b After 0 ears ( half-lives), the amount of the original substance is reduced b one-half, times: 0. g remains. c. A0 a 0 0 Substitute t 0. b After 0 ears ( half-lives), the amount of the original substance is reduced b one-half, times: 0. g remains. Skill Practice a b 0. a b 0. a b 0.. Cesium 7 is a radioactive metal with a short half-life of 0 ears. In a sample originall having g of cesium 7, the amount of cesium 7 present after t ears is given b At a b t 0 a. How much cesium 7 will be present after 0 ears? b. How much cesium 7 will be present after 90 ears? Skill Practice Answers a. 0. g b. 0. g

26 7 Chapter 0 Eponential and Logarithmic Functions Eponential functions are often used to model population growth. Suppose the initial value of a population at some time t 0 is P 0. If the rate of increase of a population is r, then after,, and ears, the new population can be found as follows: After ear: a Total a initial population b population b P 0 P 0 r P 0 r increase in a population b Factor out P 0. After ears: a Total population b a population after ear b increase in a population b P 0 r P 0 rr P 0 r P 0 rr P 0 r r Factor out P 0 r. P 0 r After ears: a Total a population increase in population b after ears b a population b P 0 r P 0 r r P 0 r P 0 r r P 0 r r Factor out P 0 r. P 0 r This pattern continues, and after t ears, the population P(t) is given b Pt P 0 r t Eample Appling an Eponential Growth Function The population of the Bahamas in 000 was estimated at 00,000 with an annual rate of increase of %. a. Find a mathematical model that relates the population of the Bahamas as a function of the number of ears after 000. b. If the annual rate of increase remains the same, use this model to predict the population of the Bahamas in the ear 00. Round to the nearest thousand. Solution: a. The initial population is P 0 00,000, and the rate of increase is r %. Pt P 0 r t Substitute P 0 00,000 and r , t 00,000.0 t Here t 0 corresponds to the ear 000.

27 Section 0. Eponential Functions 7 b. Because the initial population t 0 corresponds to the ear 000, we use t 0 to find the population in the ear 00. P0 00, ,000 Skill Practice 7. The population of Colorado in 000 was approimatel,00,000 with an annual increase of.%. a. Find a mathematical model that relates the population of Colorado as function of the number of ears after 000. b. If the annual increase remains the same, use this model to predict the population of Colorado in 00. Skill Practice Answers 7a. Pt,00,000.0 t b. Approimatel,,000 Section 0. Boost our GRADE at mathzone.com! Stud Skills Eercise. Define the ke terms. Practice Eercises Practice Problems Self-Tests NetTutor e-professors Videos a. Eponential function b. Eponential growth function c. Eponential deca function Review Eercises For Eercises 7, find the functions, using f and g as given. f. f g. g f. g f g. a g. f g 7. f b g f For Eercises 0, find the inverse function.. {(, ), (0, 0),,, (0, )} 9. e,, a, b,, 0, a, 0b, 0, f 0. a, b, c, d, e, f For Eercises, evaluate the epression without the use of a calculator Concept : Approimating Eponential Epressions with a Calculator For Eercises 9, evaluate the epression b using a calculator. Round to decimal places p

28 7 Chapter 0 Eponential and Logarithmic Functions 7. Solve for. a. 9 b. 7 c. Between what two consecutive integers must the solution to lie?. Solve for. a. b. c. Between what two consecutive integers must the solution to 0 lie? 9. Solve for. a. b. c. Between what two consecutive integers must the solution to 0 lie? 0. Solve for. a. b. c. Between what two consecutive integers must the solution to 0 lie?. For f Q R find f 0, f, f, f, and f.. For g Q R find g0, g, g, g, and g.. For h p use a calculator to find h0, h, h, hqr, and hp. Round to two decimal places.. For k QR use a calculator to find k0, k, k, kp, and kqr. Round to two decimal places.. For r find r 0, r, r, r, r, and r.. For s find s 0, s, s, s, and s. Concept : Graphs of Eponential Functions 7. How do ou determine whether the graph of f b b 7 0, b is increasing or decreasing?. For f b b 7 0, b, find f 0. Graph the functions defined in Eercises 9. Plot at least three points for each function m a f g b

29 Section 0. Eponential Functions 77. n a. h. b k g. f 7 7 Concept : Applications of Eponential Functions 7. The half-life of the element radon (Rn ) is. das. In a sample originall containing g of radon, the amount left after t das is given b At 0. t/.. (Round to two decimal places, if necessar.) a. How much radon will be present after 7. das? b. How much radon will be present after 0 das?. Nobelium, an element discovered in 9, has a half-life of 0 min under certain conditions. In a sample containing g of nobelium, the amount left after t min is given b At 0. t/0. (Round to three decimal places.) a. How much nobelium is left after min? b. How much nobelium is left after hr? 9. Once an antibiotic is introduced to bacteria, the number of bacteria decreases eponentiall. For eample, beginning with million bacteria, the amount present t das from the time penicillin is introduced is given b the function At,000,000 t/. Rounding to the nearest thousand, determine how man bacteria are present after a. das b. week c. weeks 0. Once an antibiotic is introduced to bacteria, the number of bacteria decreases eponentiall. For eample, beginning with million bacteria, the amount present t das from the time streptomcin is introduced is given b the function At,000,000 t/0. Rounding to the nearest thousand, determine how man bacteria are present after a. das b. week c. weeks

30 7 Chapter 0 Eponential and Logarithmic Functions. The population of Bangladesh was,0,000 in 00 with an annual growth rate of.%. a. Find a mathematical model that relates the population of Bangladesh as a function of the number of ears after 00. b. If the annual rate of increase remains the same, use this model to predict the population of Bangladesh in the ear 00. Round to the nearest million.. The population of Figi was,000 in 00 with an annual growth rate of.07%. a. Find a mathematical model that relates the population of Figi as a function of the number of ears after 00. b. If the annual rate of increase remains the same, use this model to predict the population of Figi in the ear 00. Round to the nearest thousand.. Suppose $000 is initiall invested in an account and the value of the account grows eponentiall. If the investment doubles in 7 ears, then the amount in the account t ears after the initial investment is given b where t is epressed in ears and At is the amount in the account. a. Find the amount in the account after ears. b. Find the amount in the account after 0 ears. c. Find A0 and A7 and interpret the answers in the contet of this problem.. Suppose $00 is initiall invested in an account and the value of the account grows eponentiall. If the investment doubles in ears, then the amount in the account t ears after the initial investment is given b where t is epressed in ears and At is the amount in the account. a. Find the amount in the account after ears. b. Find the amount in the account after 0 ears. At 000 t 7 At 00 t c. Find A0 and A and interpret the answers in the contet of this problem. Graphing Calculator Eercises For Eercises, graph the functions on our calculator to support our solutions to the indicated eercises.. f. g (see Eercise 9) (see Eercise 0) 7. m. n (see Eercise ) (see Eercise ) 9. h 0. k (see Eercise ) (see Eercise )

31 Section 0. Logarithmic Functions 79. g (see Eercise ). f (see Eercise ). The function defined b A represents the total amount A in an account ears after an initial investment of $000. a. Graph A on the window where 0 and 0 0,000. b. Use Zoom and Trace to approimate the times required for the account to reach $000, $000, and $000.. The function defined b A 00 represents the total amount A in an account ears after an initial investment of $00. a. Graph A on the window where 0 0 and 0,000. b. Use Zoom and Trace to approimate the times required for the account to reach $000, $000, and $,000. Logarithmic Functions. Definition of a Logarithmic Function Consider the following equations in which the variable is located in the eponent of an epression. In some cases the solution can be found b inspection because the constant on the right-hand side of the equation is a perfect power of the base. Equation 0 0 Solution?? The equation 0 cannot be solved b inspection. However, we might suspect that is between and. Similarl, the solution to the equation 0 is between and. To solve an eponential equation for an unknown eponent, we must use a new tpe of function called a logarithmic function. Section 0. Concepts. Definition of a Logarithmic Function. Evaluating Logarithmic Epressions. The Common Logarithmic Function. Graphs of Logarithmic Functions. Applications of the Common Logarithmic Function

32 70 Chapter 0 Eponential and Logarithmic Functions Definition of a Logarithm Function If and b are positive real numbers such that b, then log b is called the logarithmic function with base b and log b is equivalent to b Note: In the epression log b, is called the logarithm, b is called the base, and is called the argument. The epression log is equivalent to b b and indicates that the logarithm is the eponent to which b must be raised to obtain. The epression log is called the logarithmic form of the equation, and the epression b b is called the eponential form of the equation. The definition of a logarithmic function suggests a close relationship with an eponential function of the same base. In fact, a logarithmic function is the inverse of the corresponding eponential function. For eample, the following steps find the inverse of the eponential function defined b f b. f b b b log b f log b Replace f b. Interchange and. Solve for using the definition of a logarithmic function. Replace b f. Eample Converting from Logarithmic Form to Eponential Form Rewrite the logarithmic equations in eponential form. a. log b. log 0 a c. log b Solution: Logarithmic Form a. log b. c. log 0 a 000 b log 0 Eponential Form Skill Practice Rewrite the logarithmic equations in eponential form.. log 9. log 0 a. log 0 00 b Skill Practice Answers

33 Section 0. Logarithmic Functions 7. Evaluating Logarithmic Epressions Eample Evaluating Logarithmic Epressions Evaluate the logarithmic epressions. a. log b. log c. log a a 0 0,000 b b d. log e. log c Qc 7 b b R f. Solution: a. log 0 0,000 is the eponent to which 0 must be raised to obtain 0,000. log 0 0, ,000 Therefore, log 0 0,000. Q R log Let represent the value of the logarithm. Rewrite the epression in eponential form. b. log Q is the eponent to which must be raised to obtain. R log a b Therefore, log Q R. Let represent the value of the logarithm. Rewrite the epression in eponential form. c. log Q R is the eponent to which must be raised to obtain. log a b a b Let represent the value of the logarithm. Rewrite the epression in eponential form. Therefore, log Q R. d. log b b is the eponent to which b must be raised to obtain b. log b b Let represent the value of the logarithm. b b Rewrite the epression in eponential form. Therefore, log b b. e. log is the eponent to which c must be raised to obtain c 7 c Qc 7 R. log c Qc 7 R Let represent the value of the logarithm. c c 7 Rewrite the epression in eponential form. 7 Therefore, log c Qc 7 R 7.

34 7 Chapter 0 Eponential and Logarithmic Functions f. log Q R log Q R is the eponent to which must be raised to obtain. log Let represent the value of the logarithm. Rewrite the epression in eponential form. Therefore, log Q R Skill Practice Evaluate the logarithmic epressions.. log log a b. log / 7. log. log b (b 0 ) 9. log. The Common Logarithmic Function The logarithmic function with base 0 is called the common logarithmic function and is denoted b log. Notice that the base is not eplicitl written but is understood to be 0. That is, log 0 is written simpl as log. Calculator Connections On most calculators, the log ke is used to compute logarithms with base 0. For eample, we know the epression log,000,000 because 0,000,000. Use the log ke to show this result on a calculator. Eample Evaluating Common Logarithms on a Calculator Evaluate the common logarithms. Round the answers to four decimal places. a. log 0 b. log Q. 0 9 R c. log Solution: a. b. c. log 0. log Q. 0 9 R 9.9 log Skill Practice Answers Skill Practice Evaluate the common logarithms. Round answers to decimal places. 0. log 00. log (. 0 ). log (0.000)

35 Section 0. Logarithmic Functions 7. Graphs of Logarithmic Functions In Section 0. we studied the graphs of eponential functions. In this section, we will graph logarithmic functions. First, recall that f log b is the inverse of g b. Therefore, the graph of f is smmetric to the graph of g about the line, as shown in Figures 0- and 0-. b, b log b, b b, 0 < b log b, 0 < b Figure 0- Figure 0- From Figures 0- and 0-, we see that the range of b is the set of positive real numbers. As epected, the domain of its inverse, the logarithmic function log b, is also the set of positive real numbers. Therefore, the domain of the logarithmic function log b is the set of positive real numbers. Eample Graphing Logarithmic Functions Graph the functions and compare the graphs to eamine the effect of the base on the shape of the graph. a. log b. log Solution: We can write each equation in its equivalent eponential form and create a table of values (Table 0-). To simplif the calculations, choose integer values of and then solve for. log or log or 0 Choose values for. Table Solve for.

36 7 Chapter 0 Eponential and Logarithmic Functions log log Figure 0- The graphs of log and log are shown in Figure 0-. Both graphs ehibit the same general behavior, and the steepness of the curve is affected b the base. The function log requires a 0-fold increase in to increase the -value b unit. The function log requires a -fold increase in to increase the -value b unit. In addition, the following characteristics are true for both graphs. The domain is the set of real numbers such that 7 0. The range is the set of real numbers. The -ais is a vertical asmptote. Both graphs pass through the point (, 0). Skill Practice. log. Graph the functions Eample illustrates that a logarithmic function with base b 7 is an increasing function. In Eample, we see that if the base b is between 0 and, the function decreases over its entire domain. Eample Graphing a Logarithmic Function Graph log. Solution: The equation log can be written in eponential form as. B choosing several values for, we can solve for and plot the corresponding points (Table 0-). The epression log defines a decreasing logarithmic function (Figure 0-). Notice that the vertical asmptote, domain, and range are the same for both increasing and decreasing logarithmic functions. Skill Practice Answers.. log log / Skill Practice Table 0- () 0 Solve for.. Graph log /. Choose values for log / Figure 0-

37 Section 0. Logarithmic Functions 7 When graphing a logarithmic equation, it is helpful to know its domain. Eample Identifing the Domain of a Logarithmic Function Find the domain of the functions. a. f log b. g log Solution: The domain of the function log b is the set of positive real numbers. That is, the argument must be greater than zero: 7 0. a. f log The argument is. The argument of the logarithm must be greater than zero. Solve for. Divide b and reverse the inequalit sign. The domain is,. b. g log The argument is. The argument of the logarithm must be greater than zero. Solve for. 7 The domain is,. Skill Practice Find the domain of the functions.. f log 7. g log Calculator Connections The graphs of Y log and Y log are shown here and can be used to confirm the solutions to Eample. Notice that each function has a vertical asmptote at the value of where the argument equals zero. Y log( ) Y log( ) The general shape and important features of eponential and logarithmic functions are summarized as follows. Skill Practice Answers. Domain: 7,. Domain: a, b

38 7 Chapter 0 Eponential and Logarithmic Functions Graphs of Eponential and Logarithmic Functions A Summar Eponential Functions Logarithmic Functions b log b b b (0, ) 0 b (, 0) 0 b Domain:, Domain: 0, Range: 0, Range:, Horizontal asmptote: 0 Vertical asmptote: 0 Passes through (0, ) Passes through (, 0) If b 7, the function is increasing. If b 7, the function is increasing. If 0 b, the function is decreasing. If 0 b, the function is decreasing. Notice that the roles of and are interchanged for the functions b and b. Therefore, it is not surprising that the domain and range are reversed between eponential and logarithmic functions. Moreover, an eponential function passes through (0, ), whereas a logarithmic function passes through (, 0). An eponential function has a horizontal asmptote at 0, whereas a logarithmic function has a vertical asmptote at 0.. Applications of the Common Logarithmic Function Eample 7 Appling a Common Logarithm to Compute ph The ph (hdrogen potential) of a solution is defined as ph log H where H represents the concentration of hdrogen ions in moles per liter (mol/l). The ph scale ranges from 0 to. The midpoint of this range, 7, represents a neutral solution. Values below 7 are progressivel more acidic, and values above 7 are progressivel more alkaline. Based on the equation ph log H, a -unit change in ph means a 0-fold change in hdrogen ion concentration. a. Normal rain has a ph of.. However, in some areas of the northeastern United States the rainwater is more acidic. What is the ph of a rain sample for which the concentration of hdrogen ions is mol/l? b. Find the ph of household ammonia if the concentration of hdrogen ions is.0 0 mol/l. Solution: a. ph log H log Substitute H (To compare this value with a familiar substance,

39 Section 0. Logarithmic Functions 77 b. ph log H log.0 0 Substitute log 0 The ph of household ammonia is. H.0 0. Skill Practice 7. A new all-natural shampoo on the market claims its hdrogen ion concentration is. 0 7 mol/l. Use the formula ph log [H ] to calculate the ph level of the shampoo. Eample Appling Logarithmic Functions to a Memor Model One method of measuring a student s retention of material after taking a course is to retest the student at specified time intervals after the course has been completed. A student s score on a calculus test t months after completing a course in calculus is approimated b St log t where t is the time in months after completing the course and S(t) is the student s score as a percent. a. What was the student s score at t 0? b. What was the student s score after months? c. What was the student s score after ear? Solution: a. b. c. St log t S0 log 0 log 0 0 St log t S log log 7. St log t S log log 7. Substitute t 0. log 0 because 0 0. The student s score at the time the course was completed was %. Use a calculator to approimate log. The student s score dropped to 7.%. Use a calculator to approimate log. The student s score dropped to 7.%. Skill Practice Answers 7. ph.

40 7 Chapter 0 Eponential and Logarithmic Functions Skill Practice Answers a. 9 b.. c. 7. Skill Practice. The memor model for a student s score on a statistics test t months after completion of the course in statistics is approimated b S t 9 log t a. What was the student s score at the time the course was completed t 0? b. What was her score after month? c. What was the score after months? Section 0. Boost our GRADE at mathzone.com! Stud Skills Eercise Practice Eercises Practice Problems Self-Tests NetTutor e-professors Videos. Define the ke terms. a. Logarithmic function b. Logarithm c. Base of a logarithm d. Argument e. Common logarithmic function f. Domain of a logarithmic function Review Eercises. For which graph of b is 0 b? i. ii.. For which graph of b is b 7? Let f.. Let g. a. Find f, f, f 0, f, and f. a. Find g, g, g 0, g, and g. b. Graph f. b. Graph g. 7 7

41 Section 0. Logarithmic Functions 79. Let r. 7. Let s. a. Find r, r, r 0, r, and r. a. Find s, s, s 0, s, and s. b. Graph r. b. Graph s. 7 7 Concept : Definition of a Logarithmic Function. For the equation log b, identif the base, the argument, and the logarithm. 9. Rewrite the equation in eponential form. log b For Eercises 0, write the equation in logarithmic form b a e e. b 9 b a b For Eercises, write the equation in eponential form.. log. log. log log a b. log 7. log 7. log b log. log 7. log / 0. log b log Concept : Evaluating Logarithmic Epressions For Eercises 9, find the logarithms without the use of a calculator.. log 7 9. log 7. log log 9. log 0. log 7/.. log. log 9 9. log 0 0. log a b log / log 7. log 7. log. log 9. log r r a a