Download B. The Product, Quotient, and Power Properties of Logarithms

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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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