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Chapter 3: Problem 50
Use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(1 .\) Where possible, evaluate logarithmic expressions without using a calculator. $$\log x+7 \log y$$
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This group exercise involves exploring the way we grow. Group members should create a graph for the function that models the percentage of adult height attained by a boy who is \(x\) years old, \(f(x)=29+48.8 \log (x+1) .\) Let \(x=5,6\) 7..... \(15,\) find function values, and connect the resulting points with a smooth curve. Then create a graph for the function that models the percentage of adult height attained by a girl who is \(x\) years old, \(g(x)=62+35 \log (x-4)\) Let \(x=5,6,7, \ldots, 15,\) find function values, and connect the resulting points with a smooth curve. Group members should then discuss similarities and differences in the growth patterns for boys and girls based on the graphs.
Each group member should consult an almanac, newspaper, magazine, or the Internet to find data that can be modeled by exponential or logarithmic functions. Group members should select the two sets of data that are most interesting and relevant. For each set selected, find a model that best fits the data. Each group member should make one prediction based on the model and then discuss a consequence of this prediction. What factors might change the accuracy of the prediction?
Use Newton's Law of Cooling, \(T=C+\left(T_{0}-C\right) e^{k t},\) to solve this exercise. At 9: 00 A.M., a coroner arrived at the home of a person who had died. The temperature of the room was \(70^{\circ} \mathrm{F}\), and at the time of death the person had a body temperature of \(98.6^{\circ} \mathrm{F} .\) The coroner took the body's temperature at 9: 30 A.M., at which time it was \(85.6^{\circ} \mathrm{F},\) and again at 10: 00 A.M., when it was \(82.7^{\circ} \mathrm{F} .\) At what time did the person die?
Make Sense? In Exercises \(137-140,\) determine whether each statement makes sense or does not make sense, and explain your reasoning. Because the equations \(2^{x}=15\) and \(2^{x}=16\) are similar, I solved them using the same method.
Determine whether each statement makes sense or does not make sense, and explain your reasoning. I estimate that \(\log _{8} 16\) lies between 1 and 2 because \(8^{1}=8\) and \(8^{2}=64\).
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