Chapter 4

Graph \(y=\log x, y=\log (10 x),\) and \(y=\log (0.1 x)\) in the same viewing rectangle. Describe the relationship among the three graphs. What logarithmic property accounts for this relationship?

Use a graphing utility and the change-of-base property to graph \(y=\log _{3} x, y=\log _{25} x,\) and \(y=\log _{100} x\) in the same viewing rectangle. a. Which graph is on the top in the interval \((0,1) ?\) Which is on the bottom? b. Which graph is on the top in the interval \((1, \infty) ?\) Which is on the bottom? c. Generalize by writing a statement about which graph is on top. which is on the bottom, and in which intervals, using \(y=\log _{b} x\) where \(b>1\)

Logarithmic models are well suited to phenomena in which growth is initially rapid but then begins to level off. Describe something that is changing over time that can be modeled using a logarithmic function.

Graph \(f\) and \(g\) in the same viewing rectangle. Then describe the relationship of the graph of \(g\) to the graph of \(f.\) $$f(x)=\ln x, g(x)=\ln (x+3)$$

Graph \(f\) and \(g\) in the same viewing rectangle. Then describe the relationship of the graph of \(g\) to the graph of \(f.\) $$f(x)=\ln x, g(x)=\ln x+3$$

Graph \(f\) and \(g\) in the same viewing rectangle. Then describe the relationship of the graph of \(g\) to the graph of \(f.\) $$f(x)=\log x, g(x)=-\log x$$

Graph \(f\) and \(g\) in the same viewing rectangle. Then describe the relationship of the graph of \(g\) to the graph of \(f.\) $$f(x)=\log x, g(x)=\log (x-2)+1$$

Which one of the following is true? a. \(\frac{\log _{7} 49}{\log _{7} 7}=\log _{7} 49-\log _{7} 7\) b. \(\log _{b}\left(x^{3}+y^{3}\right)=3 \log _{b} x+3 \log _{b} y\) c. \(\log _{b}(x y)^{5}=\left(\log _{b} x+\log _{b} y\right)^{5}\) d. \(\ln \sqrt{2}=\frac{\ln 2}{2}\)

Students in a mathematics class took a final examination. They took equivalent forms of the exam in monthly intervals thereafter. The average score, \(f(t),\) for the group after \(t\) months was modeled by the human memory function \(f(t)=75-10 \log (t+1),\) where \(0 \leq t \leq 12\) Use a graphing utility to graph the function. Then determine how many months will elapse before the average score falls below 65 .

Graph \(f\) and \(g\) in the same viewing rectangle. a. \(f(x)=\ln (3 x), g(x)=\ln 3+\ln x\) b. \(f(x)=\log \left(5 x^{2}\right), g(x)=\log 5+\log x^{2}\) c. \(f(x)=\ln \left(2 x^{3}\right), g(x)=\ln 2+\ln x^{3}\) d. Describe what you observe in parts (a)-(c). \(\begin{array}{llllll}\text { Generalize } & \text { this } & \text { observation } & \text { by } & \text { writing } & \text { an }\end{array}\) equivalent expression for \(\log _{b}(M N),\) where \(M>0\) and \(N>0.\) e. Complete this statement: The logarithm of a product is equal to _________.

Graph each of the following functions in the same viewing rectangle and then place the functions in order from the one that increases most slowly to the one that increases most rapidly. $$y=x, y=\sqrt{x}, y=e^{x}, y=\ln x, y=x^{x}, y=x^{2}$$

Write as a single term that does not contain a logarithm: $$ e^{\ln 8 x^{4}-\ln 2 x^{2}} $$

Which one of the following is true? a. \(\frac{\log _{2} 8}{\log _{2} 4}=\frac{8}{4}\) b. \(\log (-100)=-2\) c. The domain of \(f(x)=\log _{2} x\) is \((-\infty, \infty)\) d. \(\log _{b} x\) is the exponent to which \(b\) must be raised to obtain \(x\)

If \(f(x)=\log _{\phi} x,\) show that \(\frac{f(x+h)-f(x)}{h}=\log _{b}\left(1+\frac{h}{x}\right)^{1 / h}, h \neq 0\)

Without using a calculator, find the exact value of $$\frac{\log _{3} 81-\log _{\pi} 1}{\log _{2 \sqrt{2}} 8-\log 0.001}$$

Solve for \(x: \log _{4}\left[\log _{3}\left(\log _{2} x\right)\right]=0.\)

Without using a calculator, determine which is the greater number: \(\log _{4} 60\) or \(\log _{3} 40.\)

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 \(x=1,2,3, \ldots, 12,\) find function values, and connect the resulting points with a smooth curve. Then create a function that models the percentage of adult height attained by a girl who is \(x\) years old, \(g(x)=\) \(62+35 \log (x-4) . \quad\) Let \(\quad x=5,6,7, \ldots, 15, \quad\) find function values, and connect the resulting points smooth curve. Group members should then discuss similarities and differences in the growth patterns for boys and girls based on the graphs.