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171S4.4.notebook April 07, 2010 MAT 171 4.4 Properties of Logarithms; Solving Exponential/ Logarithmic Equations A. Solving Equations Using the Fundamental Properties of Logarithms If you can not solve the logarithmic equation, convert logarithmic equation to exponential equation and solve. If you can not solve the exponential equation, convert exponential equation to logarithmic equation and solve. Mar 1510:35 AM B. The Product, Quotient, and Power Properties of Logarithms Mar 319:30 PM 1 171S4.4.notebook April 07, 2010 C. Solving Logarithmic Equations D. Applications of Logistic, Exponential, and Logarithmic Functions Solving a Logistics Equation A small business makes a new discovery and begins an aggressive advertising campaign, confident they can capture 66% of the market in a short period of time. They anticipate their market share will be modeled by the function M( t) represents the percentage after t days. Apr 58:05 PM 392/10. Solve each equation by applying fundamental properties. Round to thousandths. log x = 1.6 Apr 58:21 PM 2 171S4.4.notebook April 07, 2010 392/14. Solve each equation by applying fundamental properties. Round to thousandths. 10x = 0.024 Apr 58:21 PM 392/16. Solve each equation. Write answers in exact form and in approximate form to four decimal places. 2 3e0.4x = 7 Apr 58:21 PM 3 171S4.4.notebook April 07, 2010 392/20. Solve each equation. Write answers in exact form and in approximate form to four decimal places. 250e0.05x + 1 + 175 = 1175 Apr 58:21 PM 393/22. Solve each equation. Write answers in exact form and in approximate form to four decimal places. 15 = 8 ln (3x) + 7 Apr 58:21 PM 4 171S4.4.notebook April 07, 2010 393/26. Solve each equation. Write answers in exact form and in approximate form to four decimal places. ¾ ln (4x) 6.9 = 5.1 Apr 58:21 PM 393/28. Use properties of logarithms to write each expression as a single term. ln (x + 2) + ln (3x) Apr 58:46 PM 5 171S4.4.notebook April 07, 2010 393/36. Use properties of logarithms to write each expression as a single term. ln (x + 3) ln (x 1) Apr 58:46 PM 393/44. Use the power property of logarithms to rewrite each term as the product of a constant and a logarithmic term. log 15x3 Apr 58:48 PM 6 171S4.4.notebook April 07, 2010 393/48. Use the power property of logarithms to rewrite each term as the product of a constant and a logarithmic term. log ∛34 Apr 58:48 PM 393/52. Use the properties of logarithms to write the following expressions as a sum or difference of simple logarithmic terms. log (m2n) Apr 58:52 PM 7 171S4.4.notebook April 07, 2010 393/54. Use the properties of logarithms to write the following expressions as a sum or difference of simple logarithmic terms. ln (q ∛p) Apr 58:52 PM 393/55. Use the properties of logarithms to write the following expressions as a sum or difference of simple logarithmic terms. ln (x2 / y) Apr 58:52 PM 8 171S4.4.notebook April 07, 2010 393/56. Use the properties of logarithms to write the following expressions as a sum or difference of simple logarithmic terms. ln (m2 / n3) Apr 58:52 PM 393/62. Evaluate each expression using the changeofbase formula and either base 10 or base e. Answer in exact form and in approximate form using nine decimal places, then verify the result using the original base. log8 92 Apr 58:57 PM 9 171S4.4.notebook April 07, 2010 Apr 68:51 AM 393/64. Evaluate each expression using the changeofbase formula and either base 10 or base e. Answer in exact form and in approximate form using nine decimal places, then verify the result using the original base. log6 200 Apr 58:57 PM 10 171S4.4.notebook April 07, 2010 393/70. Use the changeofbase formula to write an equivalent function, then evaluate the function as indicated (round to four decimal places). Investigate and discuss any patterns you notice in the output values, then determine the next input that will continue the pattern. g(x) = log2 x; g(5), g(10), g(20) Apr 59:01 PM 393/71. Use the changeofbase formula to write an equivalent function, then evaluate the function as indicated (round to four decimal places). Investigate and discuss any patterns you notice in the output values, then determine the next input that will continue the pattern. h(x) = log9 x; h(2), h(4), h(8) Apr 59:01 PM 11 171S4.4.notebook April 07, 2010 393/78. Solve each equation and check your answers. log x 1 = log (x 9) Apr 59:05 PM 393/80. Solve each equation and check your answers. log (3x 13) = 2 log x Apr 59:05 PM 12 171S4.4.notebook April 07, 2010 393/84. Solve each equation using the uniqueness property of logarithms. log3 (x + 6) log3 x = log3 5 Apr 59:07 PM 393/86. Solve each equation using the uniqueness property of logarithms. ln (x 1) + ln 6 = ln (3x) Apr 59:07 PM 13 171S4.4.notebook April 07, 2010 394/90. Solve each logarithmic equation using any appropriate method. Clearly identify any extraneous roots. If there are no solutions, so state. log3 (x 4) + log3 (7) = 2 Apr 59:09 PM 394/94. Solve each logarithmic equation using any appropriate method. Clearly identify any extraneous roots. If there are no solutions, so state. log (x + 14) log x = log (x + 6) Apr 59:09 PM 14 171S4.4.notebook April 07, 2010 394/104. Solve each logarithmic equation using any appropriate method. Clearly identify any extraneous roots. If there are no solutions, so state. 6x + 2 = 3589 Apr 59:09 PM 394/108. Solve each logarithmic equation using any appropriate method. Clearly identify any extraneous roots. If there are no solutions, so state. 7x = 42x 1 Apr 59:09 PM 15 171S4.4.notebook April 07, 2010 394/116. Increasing sales: After expanding their area of operations, a manufacturer of small storage buildings believes the larger area can support sales of 40 units per month. After increasing the advertising budget and enlarging the sales force, sales are expected to grow according to the model where S( t) is the expected number of sales after t months. ( a) How many sales were being made each month, prior to the expansion? ( b) How many months until sales reach 25 units per month? Apr 59:15 PM Drug absorption: The time required for a certain percentage of a drug to be absorbed by the body depends on the drug’s absorption rate. This can be modeled by the function , where p represents the percent of the drug that remains unabsorbed ( expressed as a decimal), k is the absorption rate of the drug, and T( p) represents the elapsed time. 395/124. For a drug with an absorption rate of 5.7%, (a) find the time required (to the nearest hour) for the body to absorb 50% of the drug, and (b) find the percent of this drug (to the nearest half percent) that remains unabsorbed after 24 hr. Apr 59:18 PM 16