UNIT 4A MATHEMATICAL MODELING OF INVERSE, LOGARITHMIC, AND TRIGONOMETRIC FUNCTIONS Lesson 2: Modeling Logarithmic Functions

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1 Lesson : Modeling Logarithmic Functions Lesson A..1: Logarithmic Functions as Inverses Utah Core State Standards F BF. Warm-Up A..1 Debrief In the metric system, sound intensity is measured in watts per square meter. For eample, the average sound intensity at a rock concert is 0.1 watt per square meter. The threshold of human hearing is about 1 watt per square meter. 1. What is 0.1 watt per square meter written as an epression using a base of and an eponent? Recall that in order to rewrite 0.1 using a base of and an eponent, first move the decimal place to the right to create the whole number 1. Then determine the eponent by counting how many places the decimal was moved. To move from 0.1 to 1.0, the decimal was moved 1 unit to the right, creating an eponent of 1. Therefore, 0.1 written using base with an eponent is 1.. How much greater is the sound intensity at an average rock concert than the threshold of human hearing? To determine this, divide the intensity at the rock concert by the threshold of human hearing. 1 Find inverse functions. 1 = ( 1) ( 1) = = 11 The sound intensity of the average rock concert is 0 billion times greater than the threshold of human hearing.. Write an eponential function with a base of and a power of to represent sound intensity, I(). The simplest eponential function is given by I() =. Connection to the Lesson a. Solve an equation of the form f() = c for a simple function f that has an inverse and write an epression for the inverse. For eample, f() = or f() = ( + 1)/( 1) for 1. F LE. For eponential models, epress as a logarithm the solution to ab ct = d, where a, c, and d are numbers and the base b is,, or e; evaluate the logarithm using technology. Students will compare eponential epressions to verify the eistence of eponential functions. Students will write eponential functions based on data, and write the inverses of eponential functions as logarithmic functions. UA-10 A..1

2 Lesson : Modeling Logarithmic Functions Prerequisite Skills This lesson requires the use of the following skills: determining the dependent and independent variables in an eponential function based on data from a graph or in a problem statement recognizing that a logarithmic function is the inverse of the related eponential function by comparing their graphs on the same aes Introduction In this course, you have studied a variety of functions, such as trigonometric functions, quadratic functions, and the inverses of functions. You have worked with eponents in the past and probably realize that eponents are not always whole numbers. You may also recall that sometimes eponents contain variables. An eponential function is a function that has a variable in the eponent, such as f() = 5. The power is the result of raising a base to an eponent; is a power of since 5 =. The power is also the value of the function s logarithm, such as in the logarithmic function = log 5 f() and its eponential function, f() = 5. Like other functions, eponential functions have inverses, which are called logarithmic functions. A logarithmic function is the inverse of an eponential function. For eample, for the eponential function f() = 5, the inverse logarithmic function is = log 5 f(). If the eponential function is of the form f() = a, then the logarithmic function is of the form log a f() =. This confirms the relationship between a function f() = y and its inverse, g(y) =. This relationship can also be seen from the following graph of an eponential function, f() =, and its inverse logarithmic function, log f(). UA-11 A..1

3 Lesson : Modeling Logarithmic Functions y 8 f() = 6 y = log f() 6 8 Notice that the eponential function and its inverse logarithmic function are reflected across the line f() = (often written as y = ). For eample, this means that for the value =, the eponential function is given by f() = and its inverse logarithmic function is log f() = log ( ) =. In real-world problems, such as the sound-intensity eample in the Warm-Up, there will be situations in which the inverse function is more effectively used than the function from which the inverse is derived. Knowledge of the real-world domain of the function can help make the decision about whether the function or its inverse has more meaning. Another factor in deciding which function to work with is how simplified the epressions and numbers are for each function. Key Concepts As the graph in the Introduction shows, the eponential function and its inverse are one-to-one over their domains. The domain of the eponential function is (, + ). However, the domain of the logarithmic function is (0, + ). The range of the eponential function is (0, + ). The range of the logarithmic function is (, + ). This information provides more evidence that the logarithmic function is the inverse of the eponential function. UA-1 A..1

4 Lesson : Modeling Logarithmic Functions In the graphed eample, the value of the eponential function is 1 at = 0 because f(0) = 0 = 1. Correspondingly, the value of the inverse logarithmic function is 0 at = 1 because log (1) = 0. Eponential functions with more constants can be eplored using the properties of eponents or by looking at data tables generated by a graphing calculator. Use a graphing calculator to eplore the domain, range, and other critical points of the function and its inverse logarithmic function by looking at data tables of domain and function values. Follow the directions appropriate to your calculator model. On a TI-8/8: Step 1: Press [Y=]. Press [CLEAR] to delete any other functions stored on the screen. Step : At Y1, use your keypad to enter values for the function. Use [X, T, θ, n] for and [ ] for any eponents. Step : Press [GRAPH]. Press [WINDOW] to adjust the graph s aes. Step : Press [ND][GRAPH] to display a table of values. Look at the domain values around = 0. On a TI-Nspire: Step 1: Press [home] to display the Home screen. Step : Arrow down to the graphing icon, the second icon from the left, and press [enter]. Step : Enter the function to the right of f1() = and press [enter]. Step : To adjust the - and y-ais scales on the window, press [menu] and select : Window and then 1: Window Settings. Enter each setting as needed. Tab to OK and press [enter]. Step 5: To see a table of values, press [menu] and scroll down to : View, then 5: Show Table. Either calculator will show eponential function values that approach 0 as becomes negative and that increase as becomes positive. UA-1 A..1

5 Lesson : Modeling Logarithmic Functions To show the corresponding function values for the inverse logarithmic function, switch the - and y-values, as shown in the following table. Eponential function Logarithmic function y y Notice that the logarithmic function does not eist for negative domain values. The logarithmic function values can be verified with the data table. 1 f For eample, f() =, so = log ( ) f (0). For = 0, log = 0 log log (1) = 0 or 0 = 1. = Notice that the coefficient of in the function changes the value of the function to at = 0, and it changes the value of to when the inverse function is 0. Finally, the basic definitions and rules of eponents and logarithms will be needed in order to manipulate and calculate eponential and logarithmic functions, summarized as follows. Terms and Rules for Logarithms In a logarithmic equation, log a b = c, a is the base, b is the argument, and c is the logarithm of b to the base a. The base is the quantity that is being raised to an eponent in an eponential epression, such as a in the epression a, or the quantity that is raised to an eponent which is the value of the logarithm, such as in the equation log =. The argument is the result of raising the base of a logarithm to the power that is the value of the logarithm, so that b is the argument of the logarithm log a b = c. UA-1 A..1

6 Lesson : Modeling Logarithmic Functions You may recall the rules for working with eponents; for eample, according to the Product of Powers Property, when multiplying two eponents with the same base, keep the base and add the powers: a a y = a + y. The rules for various operations with logarithms are derived from the rules for eponents. The following table lists some eponent rules, followed by the equation and name of the related logarithmic rule. Eponent rule Related logarithm rule Logarithm rule name a a y = a + y log a ( y) = log a + log a y Product rule a a y y = a loga a a y y = log log Quotient rule (a ) y = a y log a y = y log a Power rule Another rule, the base change rule, allows for computing with logarithms other than base ; log a one form of the equation for this rule is log b a =. (Other forms will be discussed later.) log b This rule is particularly useful when working with calculators that only calculate with logarithms with bases of e (natural logarithms) and (common logarithms). The irrational number e has a value of approimately.188. A natural logarithm is a logarithm with a base of e. Natural logarithms are usually written in the form ln, which means log e. For eample, f() = ln (1 ) is understood to be the inverse of the function for the eponential function g() = 1 e. A common logarithm, on the other hand, is a logarithm with a base of. When writing a common logarithm, the is usually omitted, such that log = log. For eample, the logarithmic function f() = log ( 1) is understood to be the inverse function for the eponential function g() = 1. UA-15 A..1

7 Lesson : Modeling Logarithmic Functions Common Errors/Misconceptions incorrectly identifying the domain and range variables in an eponential function and in its inverse logarithmic function confusing the base with the power in epressing an eponential function as a logarithmic function, or vice versa misidentifying the domains of eponential functions and their inverse logarithmic functions misinterpreting the coefficients of a base and of a variable in a power in an eponential function when writing the inverse logarithmic function misapplying the rules of eponents and logarithms in rewriting eponential and logarithmic functions UA-16 A..1

8 Lesson : Modeling Logarithmic Functions Guided Practice A..1 Eample 1 Write the inverse logarithmic function of the eponential function f() = Isolate the eponential term. This is necessary in order to use the logarithmic function definition log a f() =. Divide both sides of the equation by 0.1; this is equal to multiplication by. f() = becomes f() = 0... Rewrite the result as a logarithm. f() = 0. becomes log [ f()] = 0... Isolate the eponent variable,. [ ] log [ f ( )] = 0. becomes = log f ( ).. Use the rules of logarithms to rewrite the result so that the simplest epression possible can be used to evaluate the function numerically. Use the product rule for logarithms to rewrite the epression as the sum of two logarithms: [ f ]= + f = log ( ) [log ] [log ( )] Use the power rule to rewrite the separate logarithms as eponentials: + f = + f = [log ] [log ( )] log log ( ) UA-1 A..1

9 Lesson : Modeling Logarithmic Functions Eample 5. Switch the domain and function variables to write the logarithmic inverse as a logarithmic function. This step is necessary if the logarithmic function is considered independent of the eponential function from which it was derived. log + log = f 1 ( ) The inverse of the eponential function f() = is f 1 ( ) = log + log. Derive the eponential function on which the logarithmic function g() = log 6 log 6 5 is based. 1. Switch the domain and function variables to write the logarithmic function as an eponential function. g() = log 6 log 6 5 becomes = log 6 [g()] log Use the rules of logarithms to rewrite the result so that the simplest epression possible can be found for the logarithmic function before it is converted to eponential form. = log 6 [g()] log 6 5 Eponential function from step 1 = log 6 [g()] log 6 5 Rewrite 5 as 5. = log 6 g() log 6 5 = [log 6 g() log 6 5] = log 6 g( ) 5 Apply the power rule to both logarithmic terms. Factor out from both terms. Use the quotient rule to rewrite the logarithmic terms. UA-18 A..1

10 Lesson : Modeling Logarithmic Functions. Solve for the logarithmic term and rewrite the logarithmic function as an eponential function. The equation may be easier to work with by applying the Symmetric Property of Equality so that is on the right side of the equation. g( ) log6 5 = log g( 5 ) = 6 log 6 [0. g()] = 0.5. Write the eponential function from the simplified logarithmic function by using the definition of an eponential function and its inverse. log 6 [0. g()] = 0.5 Logarithmic function 0. g() = Rewrite as the inverse. g() = Divide both sides by 0.. The logarithmic function g() = log 6 log 6 5 is based on the eponential function g() = Eample Compare the domain and range of the inverse logarithmic functions of the eponential functions f() = and g() = Isolate the eponential term in each function. This will be helpful in applying the definition of the logarithmic function for each eponential function. f() = g() = + f ( ) = g( ) = UA-19 A..1

11 Lesson : Modeling Logarithmic Functions. Write each eponential function as a logarithmic function. f ( ) f = becomes log ( ) =. g( ) = g becomes log ( ) =.. Use the rules of logarithms to simplify each logarithmic function as much as possible. Use the quotient rule for logarithms to simplify each logarithmic function. f Simplify log ( ) =. log f ( ) = Function log f() log = Apply the quotient rule. log log f() = Divide both sides by 1. log f = Rewrite as a fraction using the ( ) quotient rule. g Simplify log ( ) =. log g( ) = Function log g() log log = Apply the quotient rule. log log g() = log g = ( ) Divide by 1; simplify. Rewrite as a fraction using the quotient rule. UA-180 A..1

12 Lesson : Modeling Logarithmic Functions. Switch the and f() variables to write the inverse logarithmic functions. log f ( ) = becomes log g =. ( ) log f = ( ) becomes log g = ( ). 5. Find when f() = 0 and g() = 0. log f = ( ) = 0 log g = ( ) = 0 Rewrite each logarithm as an eponential and solve for : 0 0 = = = = 6. Describe what happens to the logarithmic function values as increases significantly. Substitute a large value of into each function, such as = 0. log f = ( ) f (0) = log log g = ( ) (0) g (0) = log (0) 1 If approaches 0 for f(), then approaches 0 for g() as approaches infinity for both functions. Refer back to step 1 to see that in the original 1 eponential function, the eponential term is, which equals. Therefore, in the logarithmic function, as approaches positive infinity, the logarithmic function values approach negative infinity. UA-181 A..1

13 Lesson : Modeling Logarithmic Functions. Describe what happens to the function values as approaches 0. This will describe what happens to the graphs of the inverse logarithmic functions as they grow closer to the y-ais. As approaches 0, the fractions in each function, and, approach infinity. For eample, f(0.001) = log = log (000), which results in an -value that is approimately, since = Describe what happens to the function values for -values that are less than 0. Recall that logarithmic functions are not defined for values of less than 0. This can be verified on the calculator by looking at a table of values for -values less than Describe how the logarithmic functions differ across their respective domains. For f(), the function domain is (0, + ) and the range is (, ). The function value is 0 at the domain value =. For g(), the function domain is (0, + ) and the range is (, ). The function value is 0 at the domain value =. Compare the domain values at which each function is equal to large and small function values: At f() =, = or = = 118,098. At g() =, = or = = 6,196. At f() =, = or =, which is about At g() =, = or =, which is about The function values of f() approach 0 at a slower rate than those of g(), whereas the function values of f() approach negative infinity at a faster rate than those of g(). UA-18 A..1

14 Lesson : Modeling Logarithmic Functions Eample Use a logarithmic function to solve the eponential equation = Rewrite the eponential function as its inverse logarithmic function. This is an alternative method to using the properties of eponents to solve the equation. = 5 becomes log 5=.. Simplify the result algebraically. log 5= Inverse logarithmic function log 5 = Multiply both sides by. = log 5 Add and subtract from both sides. = (1 log 5) Factor out. = 1 log 5 Divide both sides by 1 log 5. UA-18 A..1

15 Lesson : Modeling Logarithmic Functions. Solve the original eponential equation using the rules of eponents. This will serve as a check on the logarithmic approach used in steps 1 and. = 5 Original equation = 5 Simplify the fractional eponent. = 5 Rewrite subtracted eponents as a fraction. = Cross multiply and simplify. 5 = Simplify. 5 Simplify using the definition of logarithm and the quotient rule. = 5 Simplified eponential equation log = 5 Rewrite as a logarithm. log log 5= Apply the quotient rule. The resulting epression can be rearranged to equal, yielding the result found with the logarithmic function in step : log log 5= becomes = 1 log 5. UA-18 A..1

16 Lesson : Modeling Logarithmic Functions Eample 5 Write the domain and function value of the eponential function f() = and its inverse at a domain value of = Use a graphing or scientific calculator and the rules of eponents and logarithms to verify your results. 1. Substitute = 1.05 into the function f() = Evaluate the function using either a graphing calculator or a scientific calculator with eponentiation functionality. 0.(1.05) f (1.05) = Write the domain and function value of the inverse of the eponential function at = Switch the domain and function values of the eponential function to determine the domain and function values of the inverse logarithmic function at = The corresponding values for the inverse logarithmic function are given by the ordered pair (.05, 1.05).. Verify that the ordered pair determined in step satisfies the inverse logarithmic function. To verify that (.05, 1.05) satisfies the inverse logarithmic function, first derive the inverse logarithmic function from the original function using the definition of logarithm and the quotient and power rules. f() = Original function f ( ) 0. = log 1. Rewrite as a logarithm. = f log ( ) log 1. Apply the quotient rule. = log f ( ) log 1. Apply the power rule. (continued) UA-185 A..1

17 Lesson : Modeling Logarithmic Functions Switch and f() to write the inverse logarithmic function. f ( ) = log log 1. Substitute the value of found in step. f (.05) = log (.05) log 1. Substitute.05 for..05 f (.05) = log 1. Apply the quotient rule. f (.05) log 1.6 Apply the power rule.. Use a graphing calculator to estimate the value of the inverse logarithmic function at =.05. The process for estimating the function value will depend on the calculator used. Follow the directions specific to your calculator model. On a TI-8/8: Note: The TI-8/8 can only calculate natural (base-e) and common log a (base-) logarithms, so the base change rule log b a = must be log b used to rewrite the epression found in step before proceeding: = log 1.6 log 1.6 log Step 1: Press [Y=]. Press [CLEAR] to delete any other functions stored on the screen. Step : Use the keypad and [LOG] key to enter the values for the epression. Press [ENTER]. (continued) UA-186 A..1

18 Lesson : Modeling Logarithmic Functions On a TI-Nspire: Note: The TI-Nspire can calculate the logarithm directly without first converting it to a base- logarithm. Step 1: Press [home]. Step : Arrow down to the calculator icon, the first icon from the left, and press [enter]. Press [ctrl][clear] to create a new document or to clear any previous calculations on the current document. Step : Press [ctrl][ ] to bring up the log field. Step : Enter the argument of the logarithm into the blank subscript field. Tab to the net blank field to enter the base. Press [enter]. Step 5: To multiply the result by, press [ctrl][( )]. Then, enter the numbers and the operations using your keypad. Press [enter]. Either calculator will return a result of approimately Compare the results of steps and. Both procedures result in an ordered pair of approimately (.05, 1.06) for the inverse logarithmic function. UA-18 A..1