488 CHAPTER 4 Inverse, Exponential, and Logarithmic Functions (a) Using a graphing calculator, make a scatter diagram by plotting the point (ln D, ln P) for each planet on the xy-coordinate axes. Do the data points appear to be linear? (b) Determine a linear equation that models the data points. Graph the line and the data on the same coordinate axes. (c) Use this linear model to predict the period of the dwarf planet Pluto, to the nearest tenth, if its distance is 39.5. Compare the answer to the actual value of 248.5 yr. Use the change-of-base theorem to find an approximation to four decimal places for each logarithm. See Example 8. 79. log2 5 80. log2 9 81. log8 0.59 82. log8 0.71 83. log1/2 3 84. log1/3 2 85. logp e 86. logp22 87. log213 12 88. log219 5 89. log0.32 5 90. log0.91 8 Let u = lna and v = lnb. Write each expression in terms of u and v without using the ln function. 91. lnAb42aB 92. ln a3 b2 93. lnBa3 b5 94. lnA 23 a # b4B Concept Check Use the various properties of exponential and logarithmic functions to evaluate the expressions in parts (a)–(c). 95. Given g1x2 = ex, find (a) g1ln42 (b) g1ln522 (c) gAln1 eB. 96. Given ƒ1x2 = 3x, find (a) ƒ1log 3 22 (b) ƒ1log31ln322 (c) ƒ1log312 ln322. 97. Given ƒ1x2 = lnx, find (a) ƒ1e62 (b) ƒ1eln 32 (c) ƒ1e2 ln 32. 98. Given ƒ1x2 = log2 x, find (a) ƒ1272 (b) ƒ12log2 22 (c) ƒ122 log2 22. Work each problem. 99. Concept Check Which of the following is equivalent to 2 ln13x2 for x 70? A. ln9 + lnx B. ln6x C. ln6 + lnx D. ln9x2 100. Concept Check Which of the following is equivalent to ln14x2 - ln12x2 for x 70? A. 2 lnx B. ln2x C. ln4x ln2x D. ln2 101. The function ƒ1x2 = ln 0 x 0 plays a prominent role in calculus. Find its domain, its range, and the symmetries of its graph. 102. Consider the function ƒ1x2 = log3 0 x 0 . (a) What is the domain of this function? (b) Use a graphing calculator to graph ƒ1x2 = log3 0 x 0 in the window 3-4, 44 by 3-4, 44. (c) How might one easily misinterpret the domain of the function by merely observing the calculator graph? 78. (Modeling) Planets’ Distances from the Sun and Periods of Revolution The table contains the planets’ average distances D from the sun and their periods P of revolution around the sun in years. The distances have been normalized so that Earth is one unit away from the sun. For example, since Jupiter’s distance is 5.2, its distance from the sun is 5.2 times farther than Earth’s. Planet D P Mercury0.39 0.24 Venus 0.72 0.62 Earth 1 1 Mars 1.52 1.89 Jupiter 5.2 11.9 Saturn 9.54 29.5 Uranus 19.2 84.0 Neptune 30.1 164.8 Data from Ronan, C., The Natural History of the Universe, MacMillan Publishing Co., New York.
RkJQdWJsaXNoZXIy NjM5ODQ=