Posted: October 1st, 2022

National University of Health Sciences-MATH 204-Chapter 13 Exponential and Logarithmic Functions

832Chapter 13 Exponential and Logarithmic FunctionsIMPORTANT FACTS AND CONCEPTSEXAMPLESSection 13.6Properties for Solving Exponential and LogarithmicEquationsa) If x = y, then ax = ay.a) If x = 5, then 3x = 35.b) If ax = ay, then x = y.b) If 3x = 35, then x = 5.c) If x = y, then logb x = logb y 1x 7 0, y 7 02.c) If x = 2, then log x = log 2.d) If logb x = logb y, then x = y 1x 7 0, y 7 02.d) If log x = log 2, then x = 2.Properties 6a–6dSection 13.7Graph f1x2 = ex and g1x2 = ln x on the same set of axes.The natural exponential function isf1x2 = exwhere e L 2.7183.yNatural logarithms are logarithms to the base e. Natural logarithms are indicated by the notation ln.4y⫽x3loge x = ln xf (x) ⫽ e xFor x 7 0, if y = ln x, then e = x.y21⫺4 ⫺3 ⫺2 ⫺1⫺1⫺2The natural logarithmic function is⫺3g1x2 = ln x1234xg(x) ⫽ ln x⫺4where the base e L 2.7183.To approximate natural exponential and natural logarithmicvalues, use a scientific or graphing calculator.ln 5.83 L 1.7630If ln x = -2.09, then x = e-2.09 = 0.1237.The natural exponential function, f1x2 = ex, and the naturallogarithmic function, g1x2 = ln x, are inverses of each other.Change of Base FormulaFor any logarithm bases a and b, and positive number x,logb xloga x =logb alog5 98 =Properties for Natural Logarithmsln xy = ln x + ln yxln = ln x - ln yy1x 7 0 and y 7 02Product rule1x 7 0 and y 7 02Quotient ruleln xn = n ln x1x 7 02Power rulelog 98L 2.8488log 5ln 7 # 30 = ln 7 + ln 30lnx + 1= ln 1x + 12 - ln 1x + 82x + 8ln m5 = 5 ln mAdditional Properties for Natural Logarithms and NaturalExponential Expressionsln ex = xProperty 7ln e19 = 19eln x = x, x 7 0Property 8eln 2 = 2Chapter 13 Review Exercises[13.1] Given f1x2 = x2 - 3x + 4 and g1x2 = 2x - 5, find the following.1. 1f ⴰ g21x22. 1f ⴰ g21323. 1g ⴰ f21x24. 1g ⴰ f21-32Given f1x2 = 6x + 7 and g1x2 = 1x - 3, x Ú 3, find the following.5. 1f ⴰ g21x26. 1g ⴰ f21x205/05/2016 - RS0000000000000000000000176133 - Elementary & Intermediate Algebra for CollegeStudents, Media UpdateChapter 13 Review Exercises833Determine whether each function is a one-to-one function.7.8.y9. {16, 22, 14, 02, 1-5, 72, 13, 82}yxx110. e 10, -22, 16, 12, 13, -22, a , 4 b f212. y = x2 - 911. y = 1x + 8, x Ú - 8In Exercises 13 and 14, for each function, find the domain and range of both f1x2 and f-11x2.13. {15, 32, 16, 22, 1-4, -32, 1-1, 82}14.y4xIn Exercises 15 and 16, find f-11x2 and graph f1x2 and f-11x2 on the same axes.15. y = f1x2 = 4x - 216. y = f1x2 = 13x - 117. Yards to Feet The function f1x2 = 36x converts yards, x,into inches. Find the inverse function that converts inchesinto yards. In the inverse function, what do x and f-11x2represent?21. Compound Interest Jim Marino invested $1500 in a savingsaccount that earns 4% interest compounded quarterly. User ntthe compound interest formula A = pa1 + b to deternmine the amount in Jim’s account after 5 years.18. Gallons to Quarts The function f1x2 = 4x converts gallons,x, into quarts. Find the inverse function that converts quartsinto gallons. In the inverse function, what do x and f-11x2represent?[13.2] Graph the following functions.© Thinkstock19. y = 2x1 x20. y = a b2[13.3] Write each equation in logarithmic form.22. 82 = 64Write each equation in exponential form.25. log2 32 = 524. 5-3 =23. 811>4 = 326. log1>41= 21627. log611251= -236Write each equation in exponential form and find the missing value.28. 3 = log4 x29. 4 = loga 8130. -3 = log1>5 xGraph the following functions.31. y = log3 x32. y = log1>2 x05/05/2016 - RS0000000000000000000000176133 - Elementary & Intermediate Algebra for CollegeStudents, Media Update834Chapter 13 Exponential and Logarithmic Functions[13.4] Use the properties of logarithms to expand each expression.34. log3 1x - 933. log5 17835. log61a + 1236. log19x4712x + 325Write the following as the logarithm of a single expression.37. 5 log x - 3 log 1x + 1239.38. 41log 2 + log x2 - log y1[ln x - ln 1x + 22] - ln 2340. 3 ln x +1ln 1x + 12 - 6 ln 1x + 422Evaluate.41. 8log 8 1042. log4 4543. 11log 9 8144.log8189[13.5, 13.7] Use a calculator to approximate each logarithm. Round your answers to four decimal places.45. log 81946. ln 0.0281Use a calculator to approximate each power of 10. Round your answers to four decimal places.48. 10⫺3.15747. 103.159Use a calculator to find x. Round your answers to four decimal places.49. log x = 4.06350. log x = - 1.2262Evaluate.51. log 10552. 10log 9[13.6] Solve without using a calculator.55. 625 = 5x56. 49x =1753. 7 log 103.254. 2110log 4.7257. 23x - 1 = 3258. 27x = 32x + 561. 12.5x + 1 = 38162. 3x + 2 = 8xSolve using a calculator. Round your answers to four decimal places.59. 7x = 15260. 3.1x = 856Solve the logarithmic equation.63. log7 12x - 32 = 264. log x + log 14x - 192 = log 565. log3 x + log3 12x + 12 = 166. ln 1x + 12 - ln 1x - 22 = ln 4[13.7] Solve each exponential equation for the remaining variable. Round your answers to four decimal places.68. 100 = A0 e-0.4213267. 50 = 25e0.6tSolve for the indicated variable.69. A = A0 ekt, for t70. 200 = 800ekt, for k71. ln y - ln x = 6, for y72. ln 1y + 12 - ln 1x + 82 = ln 3, for yUse the change of base formula to evaluate. Write the answers rounded to four decimal places.73. log2 19674. log3 47[13.2–13.7]75. Compound Interest Find the amount of money accumulatedif Justine Elwood puts $12,000 in a savings account yielding 6% interest compounded annually for 8 years. UseA = pa 1 +r ntb .n76. Interest Compounded Continuously If $6000 is placed in asavings account paying 4% interest compounded continuously, find the time needed for the account to double invalue.77. Bacteria The bacteria Escherichia coli are commonly foundin the bladders of humans. Suppose that 2000 bacteria arepresent at time 0. Then the number of bacteria present t minutes later may be found by the function N1t2 = 20001220.05t.a) When will 50,000 bacteria be present?b) Suppose that a human bladder infection is classified as acondition with 120,000 bacteria. When would a persondevelop a bladder infection if he or she started with 2000bacteria?05/05/2016 - RS0000000000000000000000176133 - Elementary & Intermediate Algebra for CollegeStudents, Media Update835Chapter 13 Practice Test78. Atmospheric Pressure The atmospheric pressure, P, inpounds per square inch at an elevation of x feet above sealevel can be found by the formula P = 14.7e-0.00004x. Find theatmospheric pressure at the top of Half Dome in YosemiteNational Park, an elevation of 8842 feet.79. Retention A class of history students is given a final exam atthe end of the course. As part of a research project, thestudents are also given equivalent forms of the exameach month for n months. The average grade of theclass after n months may be found by the functionA1n2 = 72 - 18 log1n + 12, n Ú 0.a) What was the class average when the students took theoriginal exam 1n = 02?b) What was the class average for the exam given 3 monthslater?© Allen R. Angelc) After how many months was the class average 58.0?Yosemite National ParkChapter 13 Practice TestChapter Test Prep Videos provide fully worked-out solutions to any of the exercises you want to review.Chapter Test Prep Videos are available via, or on(search “Angel Elementary andIntermediate Algebra” and click on “Channels”)1. a) Determine whether the following function is a one-toone function.{14, 22, 1-3, 82, 1-1, 32, 16, -72}14. Expand log2x31x - 42x + 2.15. Write as the logarithm of a single expression.b) List the set of ordered pairs in the inverse function.2. Given f1x2 = x2 - 3 and g1x2 = x + 2, find a) 1f ⴰ g21x2.b) 1f ⴰ g2162.3. Given f1x2 = x2 + 8 and g1x2 = 1x - 5, x Ú 5, finda) 1g ⴰ f21x2. b) 1g ⴰ f2172.-1-1In Exercises 4 and 5, a) find f 1x2 and b) graph f1x2 and f 1x2on the same axes.4. y = f1x2 = -3x - 55. y = f1x2 = 1x - 1, x Ú 16. What is the domain of y = log5 x?7. Evaluate log41.2568. Graph y = 3x.9. Graph y = log2 x.10. Write 2-51=in logarithmic form.3211. Write log5 125 = 3 in exponential form.Write Exercises 12 and 13 in exponential form and find the missingvalue.12. 4 = log2 1x + 3213. y = log64 167 log6 1x - 42 + 2 log6 1x + 32 -1log6 x.216. Evaluate 10 log9 19.17. a) Find log 4620 rounded to 4 decimal places.b) Find ln 0.0692 rounded to 4 decimal places.18. Solve 3x = 19 for x.19. Solve log 4x = log 1x + 32 + log 2 for x.20. Solve log 1x + 52 - log 1x - 22 = log 6 for x.21. Find N, rounded to 4 decimal places, if ln N = 2.79.22. Evaluate log6 40, rounded to 4 decimal places, using thechange of base formula.23. Solve 100 = 250e-0.03t for t, rounded to 4 decimal places.24. Savings Account What amount of money accumulates ifKim Lee puts $3500 in a savings account yielding 4% interestcompounded quarterly for 10 years?25. Carbon 14 The amount of carbon 14 remaining after t yearsis found by the formula v = v0 e-0.0001205t, where v0 is the original amount of carbon 14. If a fossil originally contained 60grams of carbon 14, and now contains 40 grams of carbon 14,how old is the fossil?05/05/2016 - RS0000000000000000000000176133 - Elementary & Intermediate Algebra for CollegeStudents, Media Update

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