3.3 Properties of Logarithms

Transcription

1 Section 3.3 Properties of Logarithms Properties of Logarithms Change of Base Most calculators have only two types of log keys, one for common logarithms (base 0) and one for natural logarithms (base e). Although common logs and natural logs are the most frequently used, you may occasionally need to evaluate logarithms to other bases. To do this, you can use the following change-of-base formula. Change-of-Base Formula Let a, b, and be positive real numbers such that a and b. Then log a can be converted to a different base using any of the following formulas. Base b Base 0 Base e log a log b log b a log a log 0 log 0 a log a ln ln a What you should learn Rewrite logarithms with different bases. Use properties of logarithms to evaluate or rewrite logarithmic epressions. Use properties of logarithms to epand or condense logarithmic epressions. Use logarithmic functions to model and solve real-life problems. Why you should learn it Logarithmic functions can be used to model and solve real-life problems, such as the human memory model in Eercise 96 on page. One way to look at the change-of-base formula is that logarithms to base a are simply constant multiples of logarithms to base The constant multiplier is log b a. Eample a. log 4 5 log 0 5 log 0 4 Changing Bases Using Common Logarithms log a log 0 log 0 a Use a calculator. Gary Conner/PhotoEdit log log 0 log Now try Eercise 9. Eample a. log 4 5 Changing Bases Using Natural Logarithms ln 5 ln log a ln ln a Use a calculator. STUDY TIP Notice in Eamples and that the result is the same whether common logarithms or natural logarithms are used in the change-of-base formula. log ln ln Now try Eercise 5.

2 08 Chapter 3 Eponential and Logarithmic Functions Properties of Logarithms You know from the previous section that the logarithmic function with base a is the inverse function of the eponential function with base a. So, it makes sense that the properties of eponents (see Section 3.) should have corresponding properties involving logarithms. For instance, the eponential property a 0 has the corresponding logarithmic property log a 0. Properties of Logarithms (See the proof on page 55.) Let a be a positive number such that a, and let n be a real number. If u and v are positive real numbers, the following properties are true. Logarithm with Base a Natural Logarithm. :. Quotient Property: log a uv log a u log a v lnuv ln u ln v log u a v log a u log a v ln u ln u ln v v STUDY TIP There is no general property that can be used to rewrite log a u ± v. Specifically, log a y is not equal to log a log a y. 3. : log ln u n a u n n log a u n ln u Eample 3 Using Properties of Logarithms Write each logarithm in terms of ln and ln 3. a. ln 6 ln ln 6 ln 3 7 a. Rewrite 6 as 3. ln ln 3 ln ln ln 7 7 ln ln 3 3 Quotient Property Rewrite 7 as 3 3. ln 3 ln 3 Now try Eercise 7. Eample 4 Using Properties of Logarithms Use the properties of logarithms to verify that log 0 00 log 000 log 0 00 log 0 00 Now try Eercise 35. log 0 00 log Rewrite Simplify. 00 as 00.

3 Rewriting Logarithmic Epressions The properties of logarithms are useful for rewriting logarithmic epressions in forms that simplify the operations of algebra. This is true because they convert complicated products, quotients, and eponential forms into simpler sums, differences, and products, respectively. Section 3.3 Properties of Logarithms 09 Eample 5 Epanding Logarithmic Epressions Use the properties of logarithms to epand each epression. a. log y a. log log 4 5 log y log 4 y ln ln 7 log log 4 log 4 y ln ln3 5 ln 7 ln3 5 ln 7 Now try Eercise 55. Quotient Property In Eample 5, the properties of logarithms were used to epand logarithmic epressions. In Eample 6, this procedure is reversed and the properties of logarithms are used to condense logarithmic epressions. Eample 6 Condensing Logarithmic Epressions Use the properties of logarithms to condense each logarithmic epression. a. log 0 3 log 0 ln ln c. 3log log 4 a. log 0 3 log 0 log 0 log 0 3 log 0 3 Rewrite radical using rational eponent. ln ln ln ln Eploration Use a graphing utility to graph the functions and y ln ln 3 y ln 3 in the same viewing window. Does the graphing utility show the functions with the same domain? If so, should it? Eplain your reasoning. ln Quotient Property c. 3log log 4 3 log 4 log 4 3 log 3 Rewrite with a 4 radical. Now try Eercise 7.

4 0 Chapter 3 Eponential and Logarithmic Functions Eample 7 Finding a Mathematical Model The table shows the mean distance from the sun and the period y (the time it takes a planet to orbit the sun) for each of the si planets that are closest to the sun. In the table, the mean distance is given in astronomical units (where the Earth s mean distance is defined as.0), and the period is given in years. Find an equation that relates y and. Planet Mercury Venus Earth Mars Jupiter Saturn Mean distance, Period, y Period (in years) y Figure 3.9 Mercury Venus Earth Mars Saturn Jupiter Mean distance (in astronomical units) Algebraic The points in the table are plotted in Figure 3.9. From this figure it is not clear how to find an equation that relates y and. To solve this problem, take the natural log of each of the - and y-values in the table. This produces the following results. Planet Mercury Venus Earth ln X ln y Y Planet Mars Jupiter Saturn ln X ln y Y Now, by plotting the points in the table, you can see that all si of the points appear to lie in a line, as shown in Figure Choose any two points to determine the slope of the line. Using the two points 0.4, 0.63 and 0, 0, you can determine that the slope of the line is m By the point-slope form, the equation of the line is Y 3 X, where Y ln y and X ln. You can therefore conclude that ln y 3 ln. Now try Eercise 97. Graphical The points in the table are plotted in Figure 3.9. From this figure it is not clear how to find an equation that relates y and. To solve this problem, take the natural log of each of the - and y-values in the table. This produces the following results. Planet Mercury Venus Earth ln X ln y Y Planet Mars Jupiter Saturn ln X ln y Y Now, by plotting the points in the table, you can see that all si of the points appear to lie in a line, as shown in Figure Using the linear regression feature of a graphing utility, you can find a linear model for the data, as shown in Figure 3.3. You can approimate this model to be Y.5X 3 X, where Y ln y and X ln. From the model, you can see that the slope of the line 3 is So, you can conclude that ln y 3. ln. 4 4 Figure 3.30 Figure 3.3 In Eample 7, try to convert the final equation to y f form. You will get a function of the form y a b, which is called a power model.

5 Section 3.3 Properties of Logarithms 3.3 Eercises See for worked-out solutions to odd-numbered eercises. Vocabulary Check Fill in the blanks.. To evaluate logarithms to any base, you can use the formula.. The change-of-base formula for base e is given by log a. 3. n log a u 4. lnuv In Eercises 8, rewrite the logarithm as a ratio of (a) common logarithms and (b) natural logarithms.. log 5. log 3 3. log 5 4. log 3 5. log 3 a 0 6. log 3 a 4 7. log.6 8. log 7. In Eercises 9 6, evaluate the logarithm using the changeof-base formula. Round your result to three decimal places. 9. log log 7 4. log 4. log log log log log 0 35 In Eercises 7 0, rewrite the epression in terms of ln 4 and ln ln 0 8. ln In Eercises 4, approimate the logarithm using the properties of logarithms, given that log b y 0.356, log b 3 y , and log b 5 y Round your result to four decimal places.. log b 5. log b log 4. log b 5 b 3 In Eercises 5 30, use the change-of-base formula log a ln /ln a and a graphing utility to graph the function. 5. f log 3 6. f log 7. f log 8. f log 3 9. f log ln 5 64 f log ln 5 9 In Eercises 3 34, use the properties of logarithms to rewrite and simplify the logarithmic epression. 3. log log ln5e ln 6 e In Eercises 35 and 36, use the properties of logarithms to verify the equation. 35. log log ln 4 3 ln ln 3 In Eercises 37 56, use the properties of logarithms to epand the epression as a sum, difference, and/or constant multiple of logarithms. (Assume all variables are positive.) 37. log log 0 0z 39. log log y 0 4. log log 6 z ln z 44. ln t ln yz 46. ln y z 47. log 3 a bc log 5 3 y 3 z lna a, lnzz, a > z > 5. ln 3 5. y 53. ln > 54. ln 3, 55. ln 4 y 56. log y4 b z 5 ln y 3 z 4

6 Chapter 3 Eponential and Logarithmic Functions Graphical Analysis In Eercises 57 and 58, (a) use a graphing utility to graph the two equations in the same viewing window and (b) use the table feature of the graphing utility to create a table of values for each equation. (c) What do the graphs and tables suggest? Eplain your reasoning. 57. y ln 3 4, 58. In Eercises 59 76, condense the epression to the logarithm of a single quantity. 59. ln ln ln y ln z 6. log 4 z log 4 y 6. log 5 8 log 5 t 63. log log 7 z ln ln ln 67. ln 3 ln 68. ln ln 69. ln ln ln ln y 4 ln z ln ln ln 4ln z lnz 5 lnz ln 3 ln ln 74. ln ln ln 75. 3ln y ln y 4 ln y 76. ln ln 3 ln Graphical Analysis In Eercises 77 and 78, (a) use a graphing utility to graph the two equations in the same viewing window and (b) use the table feature of the graphing utility to create a table of values for each equation. (c) What do the graphs and tables suggest? Verify your conclusion algebraically. 77. y ln 8 ln, 78. y ln, y ln ln, y 3 ln ln 4 y ln ln y ln 64 y ln Think About It In Eercises 79 and 80, (a) use a graphing utility to graph the two equations in the same viewing window and (b) use the table feature of the graphing utility to create a table of values for each equation. (c) Are the epressions equivalent? Eplain. 79. y ln, y ln 80. y y ln 4 ln 4 ln 4, In Eercises 8 94, find the eact value of the logarithm without using a calculator. If this is not possible, state the reason. 8. log 8. log log 84. log log log log 5 75 log log 4 log ln e 3 ln e ln e 6 ln e 5 9. ln e 4 9. ln e ln e 94. ln e Sound Intensity The relationship between the number of decibels and the intensity of a sound I in watts per square meter is given by 0 log 0 I 0. (a) Use the properties of logarithms to write the formula in a simpler form. (b) Use a graphing utility to complete the table. I (c) Verify your answers in part (b) algebraically. 96. Human Memory Model Students participating in a psychology eperiment attended several lectures and were given an eam. Every month for the net year, the students were retested to see how much of the material they remembered. The average scores for the group are given by the human memory model f t 90 5 log 0 t, 0 t where t is the time (in months). (a) Use a graphing utility to graph the function over the specified domain. (b) What was the average score on the original eam t 0? (c) What was the average score after 6 months? (d) What was the average score after months? (e) When did the average score decrease to 75? 97. Comparing Models A cup of water at an initial temperature of 78C is placed in a room at a constant temperature of C. The temperature of the water is measured every 5 minutes during a half-hour period. The results are recorded as ordered pairs of the form t, T, where t is the time (in minutes) and T is the temperature (in degrees Celsius). 0, 78.0, 5, 66.0, 0, 57.5, 5, 5., 0, 46.3, 5, 4.5, 30, 39.6

7 Section 3.3 Properties of Logarithms 3 (a) The graph of the model for the data should be asymptotic with the graph of the temperature of the room. Subtract the room temperature from each of the temperatures in the ordered pairs. Use a graphing utility to plot the data points t, T and t, T. (b) An eponential model for the data t, T is given by Solve for T and graph the model. Compare the result with the plot of the original data. (c) Take the natural logarithms of the revised temperatures. Use a graphing utility to plot the points t, lnt and observe that the points appear linear. Use the regression feature of a graphing utility to fit a line to the data. The resulting line has the form Use the properties of logarithms to solve for T. Verify that the result is equivalent to the model in part (b). (d) Fit a rational model to the data. Take the reciprocals of the y-coordinates of the revised data points to generate the points Use a graphing utility to plot these points and observe that they appear linear. Use the regression feature of a graphing utility to fit a line to the data. The resulting line has the form Solve for T, and use a graphing utility to graph the rational function and the original data points. 98. Writing Write a short paragraph eplaining why the transformations of the data in Eercise 97 were necessary to obtain the models. Why did taking the logarithms of the temperatures lead to a linear scatter plot? Why did taking the reciprocals of the temperatures lead to a linear scatter plot? Synthesis True or False? In Eercises 99 06, determine whether the statement is true or false given that f ln, where > 0. Justify your answer. 0. f a T t. lnt at t, T. at T 99. fa fa f, a > f a f fa, > a f, fa 0 fa 0. f a ffa, a > f f 04. f n nf 05. If f < 0, then 0 < < e. 06. If f > 0, then > e. log 07. Proof Prove that a log ab log a b. 08. Think About It Use a graphing utility to graph f ln in the same viewing window. Which two functions have identical graphs? Eplain why. In Eercises 09 4, use the change-of-base formula to rewrite the logarithm as a ratio of logarithms. Then use a graphing utility to graph the ratio. 09. f log 0. f log 4. f log 3. f log 3 3. f log f log Eploration For how many integers between and 0 can the natural logarithms be approimated given that ln 0.693, ln , and ln ? Approimate these logarithms. (Do not use a calculator.) Skills Review In Eercises 6 9, simplify the epression. 4y y 7. 3y y y y y In Eercises 0 5, find all solutions of the equation. Be sure to check all your solutions ln, g, h ln ln ln