LESSON EIII.E EXPONENTS AND LOGARITHMS

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1 LESSON EIII.E EXPONENTS AND LOGARITHMS LESSON EIII.E EXPONENTS AND LOGARITHMS

2 OVERVIEW Here s what ou ll learn in this lesson: Eponential Functions a. Graphing eponential functions b. Applications of eponential functions c. Solving some eponential equations Logarithmic Functions Eponential and logarithmic functions have man useful applications in fields ranging from investment banking to medicine. The are used to measure man things ranging from the strength of an earthquake to the noise level in a recording studio. In this lesson, ou will look at some applications as ou review eponential and logarithmic functions. You will start b graphing them. You will also review the properties of eponents and logarithms, and ou will see how these properties can be used to solve eponential equations. a. Eponential and logarithmic form b. Graphing logarithmic functions c. Properties of logarithms Solving Equations a. Using a calculator to approimate common and natural logarithms b. Change of base formula c. Solving logarithmic equations d. Solving eponential equations TOPIC EIII ESSENTIALS OF ALGEBRA

3 EXPLAIN EXPONENTIAL FUNCTIONS Summar You have alread graphed linear functions and quadratic functions. In this concept ou will work with eponential functions. You will graph eponential functions, look at applications of eponential functions, and solve eponential equations. Definition of an Eponential Function Here are some eamples of eponential functions: f () = g() = h() = 7 In general, an eponential function is a function of the form = f () = b. Here, the constant b is called the base and is a positive number not equal to. The independent variable,, is the eponent. Notice that in an eponential function, the variable is the eponent. The function = is not an eponential function because the variable,, is the base. The domain of an eponential function is all real numbers. The range of an eponential function is all positive real numbers. The Graph of an Eponential Function To graph an eponential function ou can make a table of points, plot the points, and join them with a smooth curve, as ou have done for other functions. Here s a table of points for the eponential function =. The graph is shown in Figure EIII.E.. = From the graph of the eponential function =, ou can see that as increases, the graph rises rapidl. As becomes more negative, the graph gets closer to the -ais, but never becomes zero or negative. The -intercept is the point (0, ). The variable,, does not have to be a whole number. For eample, when =, f() = =. You can use our calculator to approimate.7. 9 Remember, = = =. = Figure EIII.E. LESSON EIII.E EXPONENTS AND LOGARITHMS EXPLAIN 7

4 Eponential Growth and Deca = = Figure EIII.E. Here s a table of points for the eponential function = f () =. The graph of = and the graph of = are shown together in Figure EIII.E.. = 0 Notice how the graph of = compares to the graph of = : Both graphs have -intercept (0, ). For positive values of, the graph of = rises more steepl than the graph of =. For negative values of, the graph of = is closer to the -ais than the graph of =. In general, for an eponential function = b when b >, as ou increase the value of b the graph rises more steepl for positive values of and is closer to the -ais for negative values of. For an eponential function = b when b >, as ou decrease the value of b the graph rises less steepl for positive values of and is further from the -ais for negative values of. An eponential function = b with base b > represents eponential growth. You can also write = as =. = _ Figure EIII.E. For an eponential function = b when b is between 0 and, the behavior of the graph is different. For eample, here is a table of points for = f () =. The graph is shown in Figure EIII.E.. = TOPIC EIII ESSENTIALS OF ALGEBRA

5 If ou compare this table with the table for =, ou see the same -values, but the corresponding -values have changed sign. The graph of = is the reflection of the graph of = about the -ais. You can see that the graph of = decreases as ou move from left to right. Figure EIII.E. compares the graph of = with the graph of =. An eponential function = b with base 0 < b < represents eponential deca. Applications of Eponential Functions There are two ver common applications of an eponential function. The first involves eponential growth as it has to do with the calculation of compound interest. = _ Figure EIII.E. = The function A (t ) = P ( + r ) t is an eponential function describing the total amount of mone in an account after t ears. The constant P describes the amount of mone ou initiall deposit, and the constant r, written as a decimal, describes the annual interest rate. For eample, suppose ou deposit $00 in a savings account with an annual interest rate of %, compounded once a ear. Here, the initial amount ou deposit, P, is $00. The annual interest rate, %, epressed as a decimal is r =.0. Then the total amount of mone in the account after 9 ears is given b: A (9) = 00( +.0) 9 = 00(.0) 9 00(.) Here ou can use a calculator to determine that (.0) 9.. = 7. So the total amount in the account after nine ears is approimatel $7.. The second application involves eponential deca and has to do with how a radioactive substance decreases in radioactivit over time. The function A (t ) = Pb rt is an eponential function describing the amount of radioactivit left in a substance after t ears. The constant P describes the amount of radioactivit the substance started with. The base, b, is a constant that depends on the radioactive chemical ou are studing. So does r, which is called the deca constant. The Base e There is a base that is especiall useful in applications such as radioactive deca. It is also important in calculus. This base is the irrational number e. The number e also arises naturall in man applications in science. The number e is an irrational number that lies between and and is approimatel.78. The decimal representation of e,.78..., like the decimal representation of the number π or the decimal representation of the number, does not repeat nor does it stop. LESSON EIII.E EXPONENTS AND LOGARITHMS EXPLAIN 9

6 The graph of = e is the same shape as the graph of = and the graph of =. Because the base, e, is between and, the graph of = e lies between the graphs of = and =, as shown in Figure EIII.E.. Your calculator ma work differentl. = = = e You can use a calculator with an e ke to obtain approimate answers to calculations involving e. For eample, to calculate 00e 0.7 :. Enter the eponent without its sign Press the ± ke Press the e ke Multipl b Figure EIII.E. So 00e 0.7 is approimatel 99.. Continuous Compound Interest Recall the formula A(t ) = P ( + r ) t, which is used when interest is compounded annuall. There is a formula that can be obtained from this formula, and which represents continuous compounding (compounding as frequentl as possible) of interest. This formula involves e and is given b the following: A(t ) = Pe rt As before, P is the original deposit, r is the annual interest rate epressed as a decimal, and t is the number of ears. So to calculate the effect of continuous compounding on a deposit of $00 at % interest for 9 ears ou get: Here ou can use a calculator to determine that e A = 00e 0.0(9) = 00e 0. 00(.9) = 7. So the total amount with continuous compounding is approimatel $7.. Notice that this is a larger amount than the $7. that ou got b compounding onl once each ear. Eponential Equations You have seen how to solve equations such as = 7 and + 7 = 0. Now, ou will solve eponential equations. An eponential equation is an equation that contains the variable in an eponent. Here are some eamples: 7 = t = t + 0 TOPIC EIII ESSENTIALS OF ALGEBRA

7 One wa to solve an eponential equation is to make use of the following propert of eponents: If b = b, then =. (Here, b,, and are real numbers, b > 0, and b.) You will also use some of the other properties of eponents that ou have learned. Here are steps ou can use to solve some eponential equations:. Write both sides of the equation using the same base: b = b.. Set the eponents equal to each other: =.. Finish solving. For eample, to solve t = 8 t for t :. Write both sides of the equation ( ) t = ( ) t using the same base,. (t ) = (t ) t = 9t. Set the eponents equal to each other. t = 9t. Finish solving for t. = t t = Some eponential equations cannot be solved b using this technique. For eample, in the equation = 7, the number 7 cannot easil be written using the base. In the net section, ou will solve this tpe of equation b using logarithms. So t =. Sample Problems Answers to Sample Problems. Here is the graph of the eponential function f () =. On the same set of aes graph: a. = 7 b. = c. = = a. To graph = 7, notice that the base 7 is greater than the base. So the graph rises more steepl for positive, and is closer to the -ais for negative. = 7 = LESSON EIII.E EXPONENTS AND LOGARITHMS EXPLAIN

8 Answers to Sample Problems b. c. = = = = b. To graph =, compare the base to the base. c. To graph =, notice that =. = =. The amount, A (t ), of radioactivit remaining in a radioactive substance after t ears is given in kilograms b A (t ) = 000e.00t. Find the starting amount of radioactivit and the amount remaining after 00 ears. a. To find the starting amount in kilograms, A(0) = 000e.00(0) substitute t = 0 into the formula for A. b. Simplif. = 000e 0 = 000() = 000 c. 000e.00(00) c. To find the amount in kilograms remaining after 00 ears, substitute t = 00 into the formula for A. A (00) = d d. Simplif. Round our answer to two decimal places at the end of our calculations.. Solve this eponential equation for : 9 = 7 + a. Write both sides of the ( ) = ( ) + equation using the base. b. 8, + c. 8, + d. 9 b. Simplif the eponents. c. Set the eponents equal to each other. d. Finish solving for. = = = TOPIC EIII ESSENTIALS OF ALGEBRA

9 LOGARITHMIC FUNCTIONS Summar In this concept ou will look at the inverse function of an eponential function, which is called a logarithmic function. You will graph logarithmic functions and stud their properties. Definition of a Logarithm You have alread used the inverse of a function to write the same information in two different was. For eample, ou can use the squaring function to write: = 9 You can use the square root function to write the same information as: 9 = So the squaring function is the inverse of the square root function. Similarl, the logarithmic function, = log b, is the inverse of the eponential function. Here are some eamples of equations written in the eponential form and in the corresponding logarithmic form: Eponential Form Logarithmic Form = 9 log 9 = 0 = log = = log = () = log = In general, this eponential statement: is equivalent to this logarithmic statement: b L = log b = L An logarithmic statement can be rewritten as an eponential statement and vice versa. You can use this idea to calculate some logarithms. eponent = 8 base argument logarithm log 8 = base For eample, to find log : Call this epression. = log Rewrite the statement in eponential form. = Solve the eponential equation. Write both sides using the base. = Set the eponents equal to each other. = Some logarithms cannot be evaluated this wa. In the net section, ou will approimate them using our calculator. So = log =. LESSON EIII.E EXPONENTS AND LOGARITHMS EXPLAIN

10 The Graph of a Logarithmic Function When ou switch between eponential and logarithmic form ou see that the eponential function and the logarithmic function are inverse functions. Here are the steps to confirm that these functions are inverses of each other. Remember, to find the inverse, f, of f:. Replace f () with.. Switch and.. Solve for.. Replace with f (). Start with the eponential function. Replace f () with. Switch and. Solve for. Replace with f (). f () = b = b = b log b = f () = log b Since the eponential function and the logarithmic function are inverses of each other, ou can use this to our advantage in graphing = log b. Recall that: The graph of the inverse function = f () is obtained b reflecting the graph of = f () about the line =. = Figure EIII.E. = = = log Figure EIII.E.7 Here s how to graph = log. Start with a table of values for the function =. = 9 0 The graph is shown in Figure EIII.E.. To graph the inverse function, = log, reflect the graph of = about the line =. This is shown in Figure EIII.E.7. From the graph, ou can see that as increases, the graph of = log rises slowl. As gets closer to zero, the graph of = log becomes more and more negative. The -intercept is the point (, 0). 9 The domain of the logarithm function is the positive real numbers. The range of the logarithm function is the real numbers. In general, if b >, the graph of the function = f ( ) = log b behaves like the above eample, = log. TOPIC EIII ESSENTIALS OF ALGEBRA

11 Here s another eample. To graph = log : Start with a table of values for the function =. = 0 9 The graph is shown in Figure EIII.E.8. To graph the inverse function, = log line =. This is shown in Figure EIII.E.9., reflect the graph of = about the From the graph, ou can see that as increases, the graph of = log falls slowl. As gets closer to zero, the graph of = log becomes more and more positive. The -intercept is the point (, 0). The domain of the logarithm function is the positive real numbers. The range of the logarithm function is the real numbers. 9 Figure EIII.E.8 = _ = log_ = Figure EIII.E.9 = _ In general, if 0 < b <, the graph of the function = f ( ) = log b behaves like the above eample, = log. Now ou have seen the shape of the graph of a logarithmic function when the base, b, is greater than, or when the base, b, is between 0 and. You can use this information to graph a logarithmic function directl b plotting a few points that satisf the function and joining these points with a smooth curve. Base 0 and Base e For most applications of logarithms ou will use logarithms whose base is either 0 or e. Logarithms to the base 0 are called common logarithms and are usuall written without the base 0, simpl as log rather than log 0. Logarithms to the base e are called natural logarithms and are usuall written ln rather than log e. LESSON EIII.E EXPONENTS AND LOGARITHMS EXPLAIN

12 Properties of Logarithms There are several algebraic properties of logarithms that are true for an base b. The correspond to the familiar properties of eponents. Propert Eamples log b b = log 7 7 = log e e = log b = 0 log 0 = 0 ln = 0 b log b = log = e ln 8 = 8 log b b n = n log 9 = 9 ln e = Log of a Product log b uv = log b u + log b v log 7 = log 7 + log Log of a Quotient u v log b = log b u log b v log = log 7 log Log of a Power log b u n = n log b u log 8 = log 8 v 7 log b = log b v = log b v log = log Here s an eample that uses the first four properties of logarithms to simplif an epression containing logarithms. To find the value of the epression ln e + log 8 7 log 7 : Rewrite In e as log e e and use the propert log b b =. = + log 8 7 log 7 Use the propert log b = 0. = log 7 Use the propert b log b =. = + 0 = 0 So ln e + log 8 7 log 7 = 0. Here s another eample. To find the value of the epression ln log : Rewrite In as log e and use the propert log b = 0. = 0 log Use the propert log b b n = n. = 0 = So, ln log =. TOPIC EIII ESSENTIALS OF ALGEBRA

13 Now here s an eample that uses the product and quotient properties of logarithms. To rewrite the epression log using several logarithms: Use the log of a quotient propert. = log ( )( + ) log ( + 7) Use the log of a product propert. = log ( ) + log ( + ) log ( + 7) So log = log ( ) + log ( + ) log ( + 7). Here s another eample. To simplif ln : Use the log of a quotient propert. = ln ln Use the propert log b = 0. = 0 ln So, ln ( )( + ) + 7 = ln. ( )( + ) + 7 The log of a product propert states that the logarithm of a product is the sum of the separate logarithms. The log of a quotient propert states that the logarithm of a quotient is the difference of the two separate logarithms. Be careful when using the quotient propert. You can use it this wa: log 7 = log 7 log But a quotient of logs is not the same log 7 thing: log 7 log log Here s an eample that uses the logarithm of a power propert. To simplif log 7 log : Write using eponents. = log 7 log Use the log of a power propert. = log 7 log Use the propert log b b =. = log 7 So, log 7 log = log 7. = log 7 In general, ou can simplif epressions containing logarithms using an combination of the logarithmic properties. Here s another eample: To use logarithmic properties to write the epression ln + ln ( ) ln ( + ) as a single logarithm: Use the log of a power propert. = ln + ln ( ) ln ( + ) Use the log of a product propert. = ln ( ) ln ( + ) Use the log of a quotient propert. = ln So, ln + ln ( ) ln ( + ) = ln ( ). ( + ) ( ) ( + ) LESSON EIII.E EXPONENTS AND LOGARITHMS EXPLAIN 7

14 Answers to Sample Problems Sample Problems b., 8 c., d., e.. Use the eponential form to find log. a. Call this epression. b. Rewrite in eponential form. c. Rewrite using the same base on each side. d. Set the eponents equal to each other. e. Solve for.. Graph the function = log. 8 = log 8 = = = = a. Complete the table for the = eponential function =. 0 b. 0 _ = b. Plot the points and join them with a smooth curve to graph = c. = = log _ = c. Draw the line =. Reflect the graph of = about the line = to graph = log TOPIC EIII ESSENTIALS OF ALGEBRA

15 . Use properties of logarithms to show that: e ( + ) + ln = + ln ( + ) ln ( ) ln ( ). Answers to Sample Problems a. Use the log of a quotient ln e ( + ) + propert on the left side. = ln e ( + ) ln ( + ) b. Use the log of a product propert on the first logarithm. c. Factor + in the third logarithm. d. Use the log of a product propert on the third logarithm. e. Use the propert log b b = on the first logarithm. = = = = b. ln e + ln ( + ) ln ( + ) c. ln e + ln ( + ) ln [( )( )] d. ln e + ln ( + ) ln ( ) ln ( ) e. + ln ( + ) ln ( ) ln( ) f. Use the log of a power propert on the second term. = + ln ( + ) ln ( ) ln( ) LESSON EIII.E EXPONENTS AND LOGARITHMS EXPLAIN 9

16 SOLVING EQUATIONS Summar In this concept ou will use a calculator to approimate certain logarithms and eponents, and ou will change the base of logarithms. Then ou will solve a variet of equations that contain logarithms or eponents. Calculating Common Logarithms and Eponents You have alread seen how to calculate some logarithms b switching to eponential form. For eample to find log 00: Recall that the common logarithm, log 0, is often written log. Call this epression. = log 00 Rewrite in eponential form. 0 = 00 Write both sides using base 0. 0 = 0 Set the eponents equal to each other. = So log 00 =. This method was successful because 00 could be written as an integer power of the base, 0. That is, 00 = 0. This method cannot be used to find log 0 70 because 70 is not an integer power of 0. That is, there is no integer n, such that 0 n = 70. You can approimate the common logarithm of a number on our calculator. Here are the steps ou can use on man calculators.. Enter the number.. Press the log ke. For eample, to find the common logarithm of the number 70:. Enter the number. 70. Press the log ke..809 So, log You can also reverse this process to find a number if ou know its common logarithm. 0 TOPIC EIII ESSENTIALS OF ALGEBRA

17 To find a number, given its common logarithm:. Enter the logarithm in our calculator, using the ± ke if necessar.. Press the 0 ke. For eample, here s how to find if ou know that log 0 =.:. Enter the logarithm, using. the ± ke if necessar.. Press the 0 ke..79 So, if log 0 =., then.. The functions log 0 and 0 are inverses of each other. On man calculators, the appear on the same ke. Calculating Natural Logarithms and Eponents To calculate a natural logarithm ou can use the same general steps as ou did for common logarithms and eponents, but use the ln ke instead of the log ke, and use the e ke instead of the 0 ke. Recall that the natural logarithm, log e, is often written ln. For eample, to find ln.7:. Enter the number..7. Press the ln ke So, ln Now here s how to find if ou know that ln =.9:. Enter the logarithm..9. Press the ± ke..9. Press the e ke So, if ln =.9, then 0.. The functions ln and e are inverses of each other. On man calculators, the appear on the same ke. The Change of Base Formula Most calculators have a ke for common logarithms (log) and a ke for natural logarithms (ln). In order to approimate a logarithm to an other base, ou can use the following change of base formula: log b = log c logc b (Here, b, c, and are positive numbers, b, and c.) This formula lets ou find a logarithm to the base b b choosing an other convenient base, c. Usuall ou will choose base 0 or base e for c. LESSON EIII.E EXPONENTS AND LOGARITHMS EXPLAIN

18 Here s how to approimate log 8.9 b using common logarithms and the change of base formula:. In the change of base formula, log 8.9 = let c = 0, b =, and = Find log 8.9. log Find log. log Divide. log 8.9 = So, log Now here s how to approimate log 8.9 b using natural logarithms and the change of base formula:. In the change of base formula, log 8.9 = let c = e, b =, and = Find ln 8.9. ln Find In. ln.098. Divide. log 8.9 = log log0 log 8.9 log ln 8.9 ln ln 8.9 ln Notice that ou get the same final answer whether ou use common logarithms or natural logarithms in the change of base formula So, log Equations that Contain One Logarithm Here are some eamples of equations that contain one logarithmic term: log = 7 ln =. + log 7 =. Here are some steps that help to solve equations that contain one logarithmic term:. Isolate the logarithm on one side of the equation.. Rewrite the equation in eponential form. For eample, to solve the equation log = 7:. Isolate the logarithm on one side of the equation. = log log =. Rewrite the equation in eponential form. =. Approimate using a calculator. = So, 0.. TOPIC EIII ESSENTIALS OF ALGEBRA

19 Now here s an eample where the variable,, is the base. To solve the equation. + log 7 =.:. Isolate the logarithm on one side of the equation. log 7 =. Rewrite the equation in eponential form. 7 =. Solve for. Discard an = 7 negative values. =.7 Negative values of are discarded since the base,, must be positive. So,.. Equations that Contain More Than One Logarithm To solve equations that contain more than one logarithm ou can often use properties of logarithms to combine several logarithms into a single logarithm. Here are some of the useful properties of logarithms. Propert Log of a Product log b uv = log b u + log b v Eample log 7 = log 7 + log Log of a Quotient u v log b = log b u log b v log = log 7 log 7 Log of a Power log b u n = n log b u log 8 = log 8 And here are two additional properties that ou can use. Propert Eample If log b u = log b v If ln = ln.7 then u = v then =.7 If b = b If 0 + = 0 then = then + = Here are some steps to solve equations that contain several logarithms.. Rewrite the equation with all the log terms on one side.. Use properties of logs to combine into a single log.. Rewrite the equation in eponential form.. Finish solving. Here s an eample. LESSON EIII.E EXPONENTS AND LOGARITHMS EXPLAIN

20 To solve log ( + ) = log ( ):. Rewrite the equation with log ( ) + log ( + ) = all the log terms on one side.. Use properties of logs to log [( )( + )] = combine into a single log.. Rewrite the equation in eponential form. ( )( + ) =. Finish solving. + = = 0 = = ± ±.8 To ensure that the values of ou have found are actuall solutions to the original equation, ou need to check them. The definition of log b requires that > 0. Since the original equation contains the terms log ( + ) and log ( ), ou must check that + > 0 and that > 0. Check =.8: Check =.8: Is + > 0? Is + > 0? Is.8 + > 0? Is.8 + > 0? Is.8 > 0? Yes. Is 0.7 > 0? Yes. Is > 0? Is > 0? Is (.8) > 0? Is (.8) > 0? Is. > 0? Yes. Is.9 > 0? Yes. So.8 and.8 are both solutions. If all the terms in an equation are logarithmic terms, then ou can tr these steps to solve the equation:. Combine the logs into a single log on each side.. Use the propert: if log b u = log b v then u = v.. Finish solving. Here s an eample. To solve ln 7 + ln = ln ( ):. Combine the logs into a single log on each side. ln = ln ( ). Use the propert: if log b u = log b v then u = v. =. Finish solving. 7 = = TOPIC EIII ESSENTIALS OF ALGEBRA

21 Once again ou need to check that this value of is a solution. Since the original equation contains the terms ln and ln ( ) ou must check that > 0 and > 0. 7 Check = : Is > 0? Is > 0? Is Is > 0? Is > 0? Is So, = 7 7 > 0? Yes. > 0? Yes. is a solution. Equations with a Variable in the Eponent You can use logarithmic and eponential properties to solve equations that contain a variable in an eponent. Here are two eamples of such eponential equations. = e + 8 = Here s a wa to solve = :. Take the log of both sides. log = log. Use the logarithm of a power propert. log = ( ) log. Finish solving for. Distribute on the right. log = log log Collect the -terms on one side. log log = log Factor out. (log log ) = log Solve for. = log log log Here, ou cannot easil write each side of the equation with the same base. So that s wh ou use these steps. You can approimate this answer using our calculator: Here s a wa to solve e + 8 = : (0.77)... Isolate the term with the eponent. e + = 0 e + =. Take the natural log of both sides. ln e + = ln. Use the propert log b b n = n. + = ln In step ou can use an base for our logarithm. But ou will usuall choose base 0 or base e because the are available on our calculator.. Solve for. = 0.97 You can approimate this using our calculator: 0.8. ln LESSON EIII.E EXPONENTS AND LOGARITHMS EXPLAIN

22 Answers to Sample Problems Sample Problems. Use our calculator to approimate: log.7 a. In the change of base formula, log.7= let c = e, b = and =.7. ln.7 ln b..7 c..098 b. Use our calculator to find ln.7. c. Use our calculator to find ln. ln.7 = ln = ln.7 ln. d. Calculate log.7 = ln.7 ln d... Solve for : log = log ( ) a. Move the log terms to the log + log ( ) = left side of the equation. b. log [ ( )] b. Use the logarithm of a = product propert to simplif the left side of the equation. c. ( ), d. = 0 e... c. Write in eponent form. d. Write the quadratic equation in standard form. e. Solve the equation b using the quadratic formula. Use our calculator to approimate. = or f. and must be positive. f. Check the solutions. So. is the onl solution.. Solve for : + e = 9 a. Isolate the term with e = the eponent. b. e, c. In e, In d. + ln e.. b. Divide both sides b. c. Take the natural logarithm of both sides. d. Finish solving for. e. Use our calculator to approimate. = = = TOPIC EIII ESSENTIALS OF ALGEBRA

23 HOMEWORK Homework Problems Circle the homework problems assigned to ou b the computer, then complete them below. Eplain Eponential Functions. Complete the following table for the function =.. The formula A = 00e t gives the amount of radioactivit (in milligrams) remaining in a substance after t das in a laborator eperiment. Find approimatel how much is left after das.. Solve for t : t = 8. The graphs of =, = 8 and = are shown in Figure EIII.E.0. Identif which graph represents each function. A Figure EIII.E.0 B C. The formula A = 000( +.0) 0 gives the total amount in a savings account after $000 has compounded for 0 ears at a.% annual interest rate. Use our calculator to find A.. Solve for : = 7. In each statement, circle the correct choice. a. For < 0, the graph of = is closer to/further from the -ais than the graph of =. b. For > 0 the graph of = rises less steepl/more steepl than the graph of =. 8. The formula A = Pe rt gives the total amount in a savings account after a deposit of P dollars compounds continuousl for t ears at an annual interest rate r. If ou deposit $0, which is compounded continuousl at an interest rate of.7%, how much will ou have after 0 ears? 9. Solve for t : t = 8 t + 0. Graph the function f() = e b using our calculator to find the points on the graph for the values =, 0, and. Plot the points on the graph, and then join these points with a smooth curve.. The number of printed circuit boards, N, that can be tested in one da b an assembl line worker who has das of eperience is given b this formula: N = 00 e.0 Find the number of circuit boards that can be tested b a worker with das of eperience.. Solve for t : 9 t = 7 t + LESSON EIII.E EXPONENTS AND LOGARITHMS HOMEWORK 7

24 Logarithmic Functions. Write this eponential statement in logarithmic form: = 7. Write this logarithmic statement in eponential form: log u =. Simplif: a. log 7 7 b. 8 log 8. The graph of a function = b is shown in Figure EIII.E.. Graph the function = log b on the same grid. 9. Find log Graph = log.. Use properties of logarithms to rewrite this epression as a single logarithm: ln ln ( + ) + ln ( ) ln ( + ) 7. Find log 9.. Graph = log.. Use properties of logarithms to rewrite using two logarithms: 9( ) log 7 + log ( ) Solving Equations. Use our calculator to approimate: log Solve for : + log = 7 7. Solve for : = 9 8. Use our calculator to approimate: ln 8. + ln. 9. Solve for : log (7 ) = Figure EIII.E. 7. Below is a table of values for an eponential function = b. Use this table to graph the inverse function = log b Solve for : = 0. Use our calculator to approimate: 7e.. Solve for : log + log ( + ) =. Solve for : + e = 9. Use our calculator to approimate: log Solve for : ln + ln ( + ) = 0. Solve for : + = 8 8. Use properties of logarithms to rewrite this epression using two logarithms: log ( + )( 7) 8 TOPIC EIII ESSENTIALS OF ALGEBRA

25 APPLY Practice Problems Here are some additional practice problems for ou to tr. Eponential Functions. Complete the table of values for the function = f () =. 0 =. Use our calculator to complete the table of approimate values for the function = f () = e. = e 0. The graphs of = 7 and = are shown below. Identif which graph is which. A B The compound interest formula A P ( r ) t gives the total amount of mone, A, in a retirement savings account after a deposit of P dollars has compunded annuall for t ears at an interest rate r (where r is epressed as a decimal). Find the total amount of mone, A, in our account if our initial deposit of $000 has compounded annuall at an 8.% annual interest for 0 ears.. The approimate number of bacteria, N, in a sample of contaminated water is given b the formula N = e.t where t is the number of hours since the sample was collected. Find the number, N, of bacteria that are present hours after a sample of contaminated water is collected. 7. The function A P + nt gives the approimate total amount of mone, A, in a savings account where the interest is compounded n times each ear. P represents the initial deposit, r the annual interest rate (epressed as a decimal), and t the number of ears. Use this formula to find the total amount, A, obtained from a deposit of $000 left to compound times a ear for 0 ears at a.8% annual interest rate. 8. Solve for v : 8 v v 9. Solve for : e = 0. Solve for t : t t Logarithmic Functions. Rewrite this statement in logarithmic form: 7 =. Rewrite this statement in eponential form: log n P = Q. Find log 8. e 8 r n. Graph the eponential function = f () =. LESSON EIII.E EXPONENTS AND LOGARITHMS APPLY 9

26 . Below is a table of values for the eponential function b. Write the corresponding table of values for the logarithmic function log b. b Below is the graph of the function log b. Graph the function b on the same grid. = log b Solving Equations. Use our calculator to approimate to two decimal places: log 8.. Use our calculator to approimate to two decimal places: e.7. Use the change of base formula to approimate to two decimal places: log 7.8 (Round our answer at the end of our calculations.). Solve for : log = 0.. Solve for : + ln( 8) =. Solve for : log ( ) = log ( + ) 7. Solve for : log = log ( ) log ( ) 8. Solve for : log ( ) + log ( ) = 0 9. Solve for : e = 0. Solve for : 0 + = 7. Graph the function log. 7. Simplif: log Simplif: log 8 + log + log Use properties of logarithms to rewrite the following as an epression with five terms: log ( 9)( ) ( )( 7) 0. Write as a single logarithm: log + log log 0 TOPIC EIII ESSENTIALS OF ALGEBRA

27 Practice Test EVALUATE Take this practice test to be sure that ou are prepared for the final quiz in Evaluate.. The graphs of = e, =, and = are shown in Figure EIII.E.. Identif which graph represents each function. A B C Figure EIII.E.. Use our calculator to complete the following table. Then plot the points and graph the function. = e 0. The compound interest formula A(t ) = P ( + r ) t gives the total amount in a savings account after t ears when an initial deposit of P dollars has been allowed to compound at an interest rate of r. Find the total amount in a savings account when ou deposit $00 and allow it to compound at a.% annual interest rate for ears.. Solve this equation for t : 8 t + = t. Find log.. Complete the following table of values and use it to graph the function = log. = log 7. Simplif: log + log 8 e ln 7 log 7 8. Use properties of logarithms to rewrite using four logarithms: ln ( ) ( 7) Find: log Solve for : log (7) + = 7. Solve for : log log = log log ( ). Use our calculator to approimate : + + = 8 LESSON EIII.E EXPONENTS AND LOGARITHMS EVALUATE

28 TOPIC EIII ESSENTIALS OF ALGEBRA

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