MPE Review Section III: Logarithmic & Exponential Functions


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1 MPE Review Section III: Logarithmic & Eponential Functions FUNCTIONS AND GRAPHS To specify a function y f (, one must give a collection of numbers D, called the domain of the function, and a procedure for determining eactly one number y from each number in D The procedure associated with a function can be given by a graph, by a table, by an epression or equation, or by a verbal description When the domain of a function is not eplicitly stated, it is understood to be the collection of all numbers to which the procedure for the function can be applied to obtain another number The notation f (a), where a is a number from the domain of f, denotes the number obtained by applying the procedure for f to the number a When a is not in the domain of f, we say that f (a) is not defined What is the domain of the function g(t) 4 t? The computational procedure for g can be applied to all numbers that do not require taking the square root of a negative number Therefore, the domain of g is all numbers t such that  t + Let f( and a Find f(a) or state that a is not in the domain of f The computational procedure given by the equation for f can be applied to the number a to obtain a real number Therefore, is in the domain of f and + f ( ) () 8 9 What is the domain of the function 0 Let h ( + 9 g (? (a) Let c 4 Find h(c) or state that h(c) is not defined (b) Find h() or state that  is not in the domain of h + Let g ( Find g + g() A graph in a Cartesian coordinate system specifies a function if and only if every line perpendicular to the ais intersects the graph in no more than one point When a function is specified by a graph, its  
2 domain is the projection of the graph onto the ais and its range is the projection of the graph onto the yais To find f (a) from the graph of f, locate a on the ais, find the point P where a vertical line through a intersects the graph, and, finally, locate the point f (a) where the horizontal line through P intersects the yais When a number b is given, all of the numbers a such that f (a) b can be found from the graph of f First, locate b on the vertical ais Then, find all points P where a horizontal line through b intersects the graph Finally, for each one of these points of intersection P, locate the point where the vertical line through P intersects the ais The graph of a function y g( is shown in Figure F (a) Find g() or state that is not in the domain of g (b) Find g(4) or state that 4 is not in the domain of g (c) Find all numbers a such that g(a) or state that is not in the range of g (d) Find all numbers c such that g(c) 4 or state that 4 is not in the range of g (a) Since the vertical line through the point on the ais does not intersect the graph, is not in the domain of g (b) The vertical line through the point 4 on the ais intersects the graph at the point P(4, ) The horizontal line through P intersects the yais at y Therefore, g(4) (c) The horizontal line through the point y on the yais intersects the graph at the points Q(, ) and R(, ) The vertical lines through these points meet the ais at the points  and, respectively Therefore, g() and g() (d) The horizontal line through the point y 4 does not intersect the graph, so there is no number c for which Figure F g(c) 4 The number 4 is not in the range of g 4 The graph of a function y f ( is shown in Figure F (a) Find f () or state that  is not in the domain of f (b) Find f (0) or state that 0 is not defined Figure F  
3 The graph of a function y g ( is shown in Figure F (a) Find all numbers a, such that g(a) or state that is not in the range of g (b) Find all numbers c, such that g(c) or state that is not in the range of g Figure F EXPONENTIAL FUNCTIONS Functions given by the epressions of the form f ( b, where b is a fied positive number and b, are called eponential functions The number b is called the base of the eponential function Every eponential function of this form has all real numbers as its domain and all positive real numbers as its range Eponential functions can be evaluated for integer values of by inspection or by arithmetic calculation, but for most values of, they are best evaluated with a scientific calculator The graphs of eponential functions f ( b all have yintercept, include the points (, b) and,, and are increasing when b < When b >, the negative ais is an asymptote of the graph b of f ( b and when b <, the positive ais is an asymptote Use the graphing calculator to investigate the graphs of the eponential functions Learn to recognize the graphs of eponential functions and to determine the base of an eponential function from its graph Use a scientific calculator to evaluate the eponentials and logarithms in the following problems 6 Find Find e π 8 Let f ( (4) 4 Find f 9 The graph of an eponential function y b is shown in Figure F4 What is the equation for this function? Figure F4  
4 0 The graph of an eponential function f ( b is shown in Figure F What is the equation for this function? Figure F BASIC OPERATONS ON FUNCTIONS We can add, subtract, multiply, and divide functions in much the same way as we do with real numbers Let f and g be two functions The sum f + g, the difference f g, the product fg, and the quotient f are functions whose domains are the set of all real numbers common to the domains of f g and g, and defined as follows: Sum: ( f + g)( f ( + g( Difference: ( f g)( f ( g( Product: ( fg)( f ( g( f f ( Quotient: (, provided g( 0 g g( + 4 Let f ( and g ( + (a) Find f + g (b) Find f g (c) Find fg (d) Find f g + 4 ( + )( ) + 4( + ) (a) ( f + g)( f ( + g( + + ( + )( ) ( + )( ) 4( + ) 6 6 (b) ( f g)( f ( g( + ( + )( ) (c) ( fg )( f ( g(
5 + f f ( 6 6 (d) ( ) g g( Let r( 7 and s ( + (a) Find r + s (b) Find r s (c) Find s r (d) Find rs (e) Find r (f) Find s s r COMPOSITION AND INVERSE FUNCTIONS The composition of two functions y f ( and y g ( is denoted y (f º g)( or y f (g() To evaluate (f º g)(a), first evaluate g(a) to get a number b and then evaluate f (b) to get f (b) f (g(a)) (f º g)(a) The domain of f º g is all numbers a in the domain of g such that b g(a) is in the domain of f To obtain an epression for (f º g)( from epressions for f and g, replace the variable in the epression for f ( with the epression for g( and simplify the result The domain of f º g may or may not be all of the numbers that can be substituted into the simplified epression for the composite function f º g Let g( and h( +  (a) Find (h º g)(4) (b) Find (g º h)() (a) Since g(4) 4, we have (h º g)(4) h(g(4)) h() () + () (b) Since h() () + ()   and the number b  is not in the domain of g, (g º h)() is not defined Let F( + and G(  + (a) Find an epression for the composition (F º G)( (b) Find an epression for the composition (G º F)( (a) (F º G)( F(G() G( + (  + ) (b) ( Go F)( G( F( ) [ F( ] F( + ( + ) ( + ) Let F( and G(  (a) Find (F º G)() (b) Find (G º F)(6) Let f( + and g(
6 (a) Find an epression for (g º f) ( (b) Find an epression for (f º g) ( Let G( and H ( 6 + (a) Find an epression for G(H() (b) Find an epression for H(G() Many, but not all, functions f are specified by a procedure that can be reversed to obtain a new function g When this is possible, we say that f has an inverse function and that g is the inverse function for f A function f has an inverse function if and only if every horizontal line intersects the graph of f in no more than one point When a function f has an inverse function g, the graph of g is the reflection of the graph of f through the line y The graph of a function F is shown in Figure F6 Sketch the graph of the inverse function G or state that F does not have an inverse function Figure F6 Figure F7 The Horizontal Line Test shows that F (Figure F6) has an inverse function G The graph of G (Figure F7) is the reflection of the graph of F through the diagonal line y Plot a few points on the graph of G by reflecting points on the graph of F through the line y Using these points as a guide, sketch the reflection of the graph of F through the line y and obtain the graph of G as shown in Figure F7 Each of the functions y and y is the inverse of the other Construct the graphs of these two functions on the same coordinate system, either manually or by using a graphics calculator Observe that the graph of each is the reflection of the graph of the other through the diagonal line y If you use a graphing calculator, be sure to establish a coordinate system in which the units of distance on the two aes are the same  6 
7 Sketch the graph of the function f ( Determine whether f has an inverse function by eamining its graph + 6 Sketch the graph of the function F( Determine whether F has an inverse function by eamining its graph 7 Sketch the graph of the function G(  Determine whether G has an inverse function by eamining its graph The procedure for the inverse function G of a function F is the reverse of the procedure for F Intuitively, then, if we begin with a number in the domain of F, apply the procedure for F to and obtain a number w, and then apply the procedure for G (which is the reverse of the procedure for F) to this number w, we should be back at the number again Indeed, two functions f and g are a pair of inverse functions if and only if for every in the domain of f, (g º f) ( g(f () and for every in the domain of g, (f º g)( f (g() This fact can be used to determine algebraically whether a given pair of functions are inverses Determine whether f( compositions and g( Since f ( g( ) g( + ( + ) + f ( + ( ) and g ( f ( ) +, f ( the functions f and g are a pair of inverse functions Determine whether P( compositions and Q( are inverse functions by computing their are inverse functions by computing their Q( ( ) ( ) Since P( Q( ) +, Q( ( ) ( ) + the functions P and Q are not a pair of inverse functions Determine whether the following pairs of functions are inverses by computing their compositions 8 8 h ( and k(
8 F( and G ( 4 + g ( and p ( + 8 r ( and + s ( + 8 When a function y f ( has an inverse function y g(, an equation for the inverse function can, in principle, be found by solving the equation f ( y) In practice, however, it may not be possible to solve this equation by familiar methods + 4 Find an equation for the inverse function for F( 7 Sketch the graph of F, preferably on a graphing calculator, and verify that F has an inverse + 4 function Since F is given by the equation y, we can find an equation for its inverse 7 function G by solving the equation y + 4 y 7 for y We have y 7 y + 4 y y ( ) y y The inverse function for F is G( Verify from their graphs that each of the following functions has an inverse function Find an equation for each of the inverse functions y 9 4 S( y + LOGARITHMIC FUNCTIONS Since horizontal lines intersect the graphs of eponential functions f ( b (b > 0, b ) in no more than one point, every eponential function has an inverse function The inverse of the eponential  8 
9 function with base b, f ( b, is called the logarithmic function base b and is denoted log b The logarithmic function base e is called the natural logarithm function and is denoted ln( The logarithmic function base 0 is called the common logarithm function and is denoted log( These logarithmic functions can be evaluated directly by using a scientific calculator Let f ( ln( Find f (49) 6 Let f ( log( Find f ln Calculate log Since the graph of y log b is the reflection of the graph of y b through the line y, properties of the graphs of the logarithmic functions can be inferred from properties of the graphs of the eponential functions The graphs of logarithmic functions f ( log b all have intercept, include the points (b, ) and,, and are increasing when b > and decreasing when b < b When b >, the negative yais is an asymptote of the graph of f ( log b and when b <, the positive yais is an asymptote The graphs of the natural logarithm function and the common logarithm function can be generated easily on a graphing calculator Some ingenuity is required to generate the graphs of other logarithmic functions on a graphing calculator Learn to recognize the graphs of logarithmic functions and to determine the base of a logarithmic function from its graph 8 The graph of a logarithmic function is shown 9 The graph of a logarithmic function is shown in Figure F8 Find the equation for this in Figure F9 Find the equation for this function function Figure F8 Figure F99 
10 Since the eponential function f ( b and the logarithmic function g( log b are a pair of inverse functions, f (a) c if and only if g(c) a Therefore, the equations b a c and log b c a (b > 0, b ) epress the same relationship among the numbers a, b and c Rewrite the eponential equation 7 w z as a logarithmic equation The equivalent logarithmic equation is w log 7 z Rewrite the logarithmic equation log w (+ ) The equivalent eponential equation is w ( + ) y 0 Rewrite s 4 as a logarithmic equation Rewrite log b a as an eponential equation Rewrite a ( + k t ) S + R as a logarithmic equation Rewrite log p (4 + ) y as an eponential equation y as an eponential equation Some problems involving logarithmic functions can be readily solved by rewriting them in eponential form Solve the equation log 7 09 Rewrite the logarithmic equation as the eponential equation (7) 09 Use a scientific calculator to find 69 (rounded to three decimal places) Solve the equation log 8 4 Rewrite the logarithmic equation as the eponential equation 4 8 Apply the fourth root function to both sides of the equation to find 8 4 Solve the equation log 9 7 Rewrite the logarithmic equation as the eponential equation 9 7 Rewrite this eponential equation as, so each member of the equation is a power of the same base Equate the eponents and solve for to find / Solve each of the following equations 4 log ( )  0 
11 log 6 log 8 8 The logarithmic functions satisfy the identities log b AB log b A + log b B (the Product Identity), A logb logb A logb B (the Quotient Identity) and B log b A p p log b A (the Power Identity) where A and B may be positive numbers or variables, algebraic epressions or functions that take on positive values These identities are used to write epressions involving logarithmic functions in different forms to suit different purposes Write the epression log ( + 4)  4 log (  ) + log (  4) as a single logarithm and simplify Use the identities for logarithmic functions to rewrite this epression: log 4 ( 4) [ log ] ( ) ( + 4) log ( ) + log ( + 4) log + log ( 4) ( + 4) 4log ( ) + log 4 ( ) ( + 4) log ( ) Now simplify by factoring and dividing out common factors: 4 ( 4) ( + 4) + 4) 4log ( ) + log ( 4) ( 4) log log ( log 4 ( ) ( + 4)( + ) log ( ) In the following problems, write the epression as a single logarithm and simplify 7 log u + log v log (w) 4 8 log 7 (s + )  log 7 s  log 7 (s  s  ) 9 log( + 8y + y ) 4log( + 6y) log( + y) 4 ( + 4)( + ) ( ) 4 ( ) In the following problems, write the epression as a sum of multiples of logarithms  
12 40 log 8 64 w u ( + + ) 4 ln v ( + ) Since the eponential function f ( b and the logarithmic function g( log b of the same base b (b > 0, b ) are inverse functions, eponential and logarithmic functions satisfy the Composition Identities For every positive number, ( f g b log b ( ( )) and, for every number, g( f () log b (b ) The Composition Identities also hold when is replaced by any variable, algebraic epression or function that takes on values for which the compositions are defined These identities are used to simplify epressions involving logarithms and eponential functions of the same base [ log7 ( s r) + log7 ( s+ r) ] Simplify 7 Using the identities for logarithmic, write the eponent as a single logarithm The epression becomes [ log7 ( s r) + log7 ( s+ r) ] log7 ( s r)( s+ r) 7 7 Use a Composition Identity to simplify further and get Simplify ( a + b) log 4ab [ log ] 7 7 ( s r) + log7 ( s+ r) log7 ( s r)( s+ r) 7 ( s r)( s + r ) ( a+ b) [ ] [ ] ( a+ b) 4ab a ab+ b ( a b) [ ] ( a+ b) ( a b) 4ab By a Composition Identity, ( a + b) log ( a b) ( a + b) ( a b) The epression simplifies to ( a + b)( a b)  
13 Simplify the following epressions 4 [ log ( t t+ 6) log ( t 9) ( t + ) ] 4 ( t u)0 [ ( ) log t u + log( t + ut+ u ) log( t u) ] 44 (a b) (4b a) ( a + b) log [ 7 7 ] 7 4 ( st+ t [ ) ( s st ) log 4 st log 0 0 ] + 4 SOLVING EQUATIONS Many equations involving logarithmic functions can be solved by rewriting the equation so it has one logarithmic term on the left and a constant on the right, rewriting this equation as an equivalent eponential equation, and, finally, solving the eponential equation Since this procedure can introduce etraneous solutions, all solutions to the eponential equation must be checked by substitution to determine whether they satisfy the original equation Solve log (  ) + log ( + ) Use the Product Identity for Logarithms to write the equation as log (  )( + ) Rewrite this equation as the eponential equation (  )( + ), or The solutions to this quadratic equation are and  On checking by substitution in the original equation, we find that is a solution and  is not a solution Solve (ln + ln Use the Power Identity to rewrite this equation as (ln + ln Write this equation in factored form as (ln + 4)(ln  ) 0 In order for this product to be zero, we must have either ln 4 or ln Rewrite these two equations in eponential form and find that we must have either  
14 or e , e Check by substituting into the original equation and verify that both of these numbers are solutions Solve the following equations 46 log (  4) + log ( + ) log ( ) log8( + ) 8 48 log (log ( + 7)) (log 4 + log 4 Many equations involving eponentials can be solved by rewriting the equation so it has one eponential term on the left and either a constant or another eponential term on the right, applying a logarithmic function to both sides of the equation, and, finally, simplifying and solving the resulting equation ( ) ( ) Solve + 0 Round off the answer to four decimal places Rewrite the equation as ( ) ( + ) Apply the function ln or log (any logarithm function would do, but these are on the calculator) to both sides of the equation to find Simplify this equation as ( ) ( + ) [ ] ln ln [ ] Now, solve this equation for to find and, rounded to four decimal places, Solve 4( ) ( ) ln ln + ( + ) ln ( ln ln ) ln + ln + ln, ln + ln + ln 7 ln ln Round off the answer to four decimal places Multiply each term of the equation by and rewrite the equation as 4 ( ) or ( ) ( )
15 Write this equation in factored form as ( + )( 4) 0 In order for this product to be zero, we must have either or 4 From the equation we find no solution, since for every real number, > 0 Apply a logarithmic functions to both members of the equation 4 and find log log 4 Thus, rounded to four decimal places log log Solve the following equations 0 ( + ) 08 7 (8  )  e  e  9 e (4 + ) (  7) ( + ) 4 ( ) 7 MATHEMATICAL MODELS A mathematical equation which relates the variables involved in a phenomenon or situation is called a mathematical model for the situation Questions about a phenomenon or situation can often be answered from a mathematical model The weight in pounds of a certain snowball rolling down a hill is given by W e 0006S, where S is the distance (in feet) the ball has rolled How much does the ball weigh initially (when S 0)? How much does the ball weigh after it has rolled 000 feet? When S 0, W e 0 pounds When S 000, W e 6 0 pounds (rounded) Nitrogen pentoide is a solid that decomposes into the gases nitrogen dioide and oygen The function N( t) 7e (0000) t, where t is the time in seconds and N is the number of grams of nitrogen pentoide remaining in the sample at time t, is a mathematical model for the decomposition of a sample of 7 grams of nitrogen pentoide (a) According to this model, how much nitrogen pentoide remains after 000 seconds? (b) How long does this model predict it will take for grams of nitrogen pentoide to decompose from the sample (so grams remain)?  
16 6 The human ear is sensitive to sounds over a wide range of intensities Sounds of intensity 0  watts per square meter are at the threshold of hearing Sounds of intensity watt per meter cause pain for most people Because of this wide range of intensities and because the sensation of loudness seems to increase as a logarithm of intensity, the loudness D (in decibles) of a sound of intensity I is defined by the equation D log I (a) What is the loudness of a sound intensity I 0  watts per square meter, which is the threshold of human hearing? (b) Constant eposure to sound of 90 decibles (or greater) endangers hearing What is the intensity of sounds of loudness D 90 decibles? (c) A large rocket engine generates sounds of loudness 80 decibles What is the intensity of the sound generated by such an engine? (d) How loud (in decibles) is the sound of a rock band that produces sounds of intensity 08 watts per square meter? 7 The concentration C of medication in the bloodstream of a patient t hours after 0 milligrams of the medication is administered orally is given by the equation 0t t ( e e ) C 66 What is the concentration of the medication in the patient's bloodstream 4 hours after the medication is administered? FUNCTIONS AND GRAPHS { < or < } III LOGARITHMIC AND EXPONENTIAL FUNCTIONS ANSWER SECTION (a) 6 h ( 4) (b)  is not in the domain of h 4 (a) f() 0 (a) is not in the range of g (b) f(0)  (b)  and EXPONENTIAL FUNCTIONS (rounded) (rounded) (rounded) 9 y 0 f ( 4 BASIC OPERATONS ON FUNCTIONS (a) (d) ( r + s)( (b) + 7 ( rs )( (e) ( r s)( (c) + r 7 + ( s (f) 4 ( s r)( + s ( r
17 COMPOSITIONS AND INVERSE FUNCTIONS (a) undefined (a) ( go f )( (b) 7 (b) ( g) 4 (a) + G( H ( ) 4 f o ( (b) H ( G( ) f has an inverse function 6 F has an inverse function 7 G does not have an inverse function 8 are inverses 9 are not inverses 0 are not inverses are inverses y 8 T ( 4 y LOGARITHMIC FUNCTIONS 0899 (rounded) (rounded) (rounded) 8 y log 9 y log 0 4 log s b a log a (S + R) + kt p y (rounded) (rounded) u v log 8w (s + ) ( + y) log7 9 log 40 log 8 u + log8 v + log8 w + s ( s ) + 6y ln + + ln( + ) + ln 4 t  4 ( t u) ( t + u) 9 4 ( ) 44 4 t + 7st + s SOLVING EQUATIONS no solution or (rounded) (rounded) (rounded) (rounded) (rounded) MATHEMATICAL MODELS (a) 0 grams 6 (a) 0 decibels 7 C 89 mg/ml (b) 4,80 seconds (b) 0  watts/square meter (c) 0 6 watts/square meter (d) 9 decibels 4
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