Chapter 4

The number \(V\) of varieties of suburban nondomesticated wildlife in a community is approximated by the model \(V=15 \cdot 10^{0.02 x}, \quad 0 \leq x \leq 36\) where \(x\) is the number of months since the development of the community was completed. Use this model to approximate the number of months since the development was completed when \(V=50\).

Find a logarithmic equation that relates \(y\) and \(x\). Explain the steps used to find the equation.$$ \begin{array}{|l|l|l|l|l|l|l|} \hline x & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline y & 1 & 1.189 & 1.316 & 1.414 & 1.495 & 1.565 \\ \hline \end{array} $$

The number \(A\) of varieties of native prairie grasses per acre within a farming region is approximated by the model \(A=10.5 \cdot 10^{0.04 x}, \quad 0 \leq x \leq 24\) where \(x\) is the number of months since the farming region was plowed. Use this model to approximate the number of months since the region was plowed using a test acre for which \(A=70\)

Find a logarithmic equation that relates \(y\) and \(x\). Explain the steps used to find the equation.$$ \begin{array}{|l|l|l|l|l|l|l|} \hline x & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline y & 1 & 1.587 & 2.080 & 2.520 & 2.924 & 3.302 \\ \hline \end{array} $$

The demand function for a special limited edition coin set is given by \(p=1000\left(1-\frac{5}{5+e^{-0.001 x}}\right)\) (a) Find the demand \(x\) for a price of \(p=\$ 139.50\). (b) Find the demand \(x\) for a price of \(p=\$ 99.99\). (c) Use a graphing utility to confirm graphically the results found in parts (a) and (b).

Find a logarithmic equation that relates \(y\) and \(x\). Explain the steps used to find the equation.$$ \begin{array}{|l|l|l|l|l|l|l|} \hline x & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline y & 2.5 & 2.102 & 1.9 & 1.768 & 1.672 & 1.597 \\ \hline \end{array} $$

The demand function for a hot tub spa is given by \(p=105,000\left(1-\frac{3}{3+e^{-0.002 x}}\right)\) (a) Find the demand \(x\) for a price of \(p=\$ 25,000\). (b) Find the demand \(x\) for a price of \(p=\$ 21,000\). (c) Use a graphing utility to confirm graphically the results found in parts (a) and (b).

Find a logarithmic equation that relates \(y\) and \(x\). Explain the steps used to find the equation.$$ \begin{array}{|l|l|l|l|l|l|l|} \hline x & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline y & 0.5 & 2.828 & 7.794 & 16 & 27.951 & 44.091 \\ \hline \end{array} $$

The yield \(V\) (in millions of cubic feet per acre) for a forest at age \(t\) years is given by \(V=6.7 e^{-48.1 / t}, \quad t>0\) (a) Use a graphing utility to find the time necessary to obtain a yield of \(1.3\) million cubic feet per acre. (b) Use a graphing utility to find the time necessary to obtain a yield of 2 million cubic feet per acre.

The approximate lengths and diameters (in inches) of common nails are shown in the table. Find a logarithmic equation that relates the diameter \(y\) of a common nail to its length \(x\).$$ \begin{array}{|c|c|} \hline \begin{array}{c} \text { Length, } \\ x \end{array} & \begin{array}{c} \text { Diameter, } \\ y \end{array} \\ \hline 1 & 0.070 \\ \hline 2 & 0.111 \\ \hline 3 & 0.146 \\ \hline \end{array} $$$$ \begin{array}{|c|c|} \hline \begin{array}{c} \text { Length, } \\ x \end{array} & \begin{array}{c} \text { Diameter, } \\ y \end{array} \\ \hline 4 & 0.176 \\ \hline 5 & 0.204 \\ \hline 6 & 0.231 \\ \hline \end{array} $$

In a group project on learning theory, a mathematical model for the percent \(P\) (in decimal form) of correct responses after \(n\) trials was found to be \(P=\frac{0.98}{1+e^{-0.3 n}}, \quad n \geq 0\) (a) After how many trials will \(80 \%\) of the responses be correct? (That is, for what value of \(n\) will \(P=0.8\) ?) (b) Use a graphing utility to graph the memory model and confirm the result found in part (a). (c) Write a paragraph describing the memory model.

The value \(y\) (in billions of dollars) of U.S. currency in circulation (outside the U.S. Treasury and not held by banks) from 1996 to 2005 can be approximated by the model \(y=-302+374 \ln t, \quad 6 \leq t \leq 15\) where \(t\) represents the year, with \(t=6\) corresponding to 1996\. (Source: Board of Governors of the Federal Reserve System) (a) Use a graphing utility to graph the model. (b) Use a graphing utility to estimate the year when the value of U.S. currency in circulation exceeded \(\$ 600\) billion. (c) Verify your answer to part (b) algebraically.

The average monthly sales \(y\) (in billions of dollars) in retail trade in the United States from 1996 to 2005 can be approximated by the model \(y=-22+117 \ln t, \quad 6 \leq t \leq 15\) where \(t\) represents the year, with \(t=6\) corresponding to 1996\. (Source: U.S. Council of Economic Advisors) (a) Use a graphing utility to graph the model. (b) Use a graphing utility to estimate the year in which the average monthly sales first exceeded \(\$ 270\) billion. (c) Verify your answer to part (b) algebraically.

Automobiles are designed with crumple zones that help protect their occupants in crashes. The crumple zones allow the occupants to move short distances when the automobiles come to abrupt stops. The greater the distance moved, the fewer g's the crash victims experience. (One \(\mathrm{g}\) is equal to the acceleration due to gravity. For very short periods of time, humans have withstood as much as 40 g's.) In crash tests with vehicles moving at 90 kilometers per hour, analysts measured the numbers of g's experienced during deceleration by crash dummies that were permitted to move \(x\) meters during impact. The data are shown in the table. $$ \begin{array}{|l|l|l|l|l|l|} \hline x & 0.2 & 0.4 & 0.6 & 0.8 & 1.0 \\ \hline g \text { 's } & 158 & 80 & 53 & 40 & 32 \\ \hline \end{array} $$A model for these data is given by \(y=-3.00+11.88 \ln x+\frac{36.94}{x}\) where \(y\) is the number of g's. (a) Complete the table using the model.$$ \begin{array}{|l|l|l|l|l|l|} \hline x & 0.2 & 0.4 & 0.6 & 0.8 & 1.0 \\ \hline y & & & & & \\ \hline \end{array} $$(b) Use a graphing utility to graph the data points and the model in the same viewing window. How do they compare? (c) Use the model to estimate the least distance traveled during impact for which the passenger does not experience more than \(30 \mathrm{~g}\) 's. (d) Do you think it is practical to lower the number of g's experienced during impact to fewer than 23 ? Explain your reasoning.

Graphical Analysis Use a graphing utility to graph \(f(x)=\ln 5 x\) and \(g(x)=\ln 5+\ln x\) in the same viewing window. What do you observe about the two graphs? What property of logarithms is being demonstrated graphically?

Graphical Analysis You are helping another student learn the properties of logarithms. How would you use a graphing utility to demonstrate to this student the logarithmic property \(\log _{a} u^{v}=v \log _{a} u\) \((u\) is a positive number, \(v\) is a real number, and \(a\) is a positive number such that \(a \neq 1) ?\) What two functions could you use? Briefly describe your explanation of this property using these functions and their graphs.

The logarithm of the product of two numbers is equal to the sum of the logarithms of the numbers.

Find a logarithmic equation that relates \(y\) and \(x\). Explain the steps used to find the equation.Reasoning An algebra student claims that the following is true: \(\log _{a} \frac{x}{y}=\frac{\log _{a} x}{\log _{a} y}=\log _{a} x-\log _{a} y\) Discuss how you would use a graphing utility to demonstrate that this claim is not true. Describe how to demonstrate the actual property of logarithms that is hidden in this faulty claim.

Reasoning A classmate claims that the following is true: \(\ln (x+y)=\ln x+\ln y=\ln x y\) Discuss how you would use a graphing utility to demonstrate that this claim is not true. Describe how to demonstrate the actual property of logarithms that is hidden in this faulty claim.

Complete the proof of the logarithmic property \(\log _{a} u v=\log _{a} u+\log _{a} v\) Let \(\log _{a} u=x\) and \(\log _{a} v=y\). \(a^{x}=\quad\) and \(a^{y}=\quad\) Rewrite in exponential form. \(u \cdot v=\quad \cdot \quad=a \quad\) Multiply and substitute for \(u\) and \(v\). \(=x+y\) Rewrite in logarithmic form. \(\log _{a} u v=\quad+\) Substitute for \(x\) and \(y\).

The logarithm of the quotient of two numbers is equal to the difference of the logarithms of the numbers.

Complete the proof of the logarithmic property \(\log _{a} \frac{u}{v}=\log _{a} u-\log _{a} v\) Let \(\log _{a} u=x\) and \(\log _{a} v=y\). \(a^{x}=\quad\) and \(\quad a^{y}=\quad\) Rewrite in exponential form. \(\frac{u}{v}=\frac{\underline{\phantom{xx}}}{u} \quad \begin{aligned}&\text { Divide and substitute for } \\\&u \text { and } v .\end{aligned}\) \(=x-y \quad\) Rewrite in logarithmic form. \(\log _{a} \frac{u}{v}=\quad-\) Substitute for \(x\) and \(y\).

Is it possible for a logarithmic equation to have more than one extraneous solution? Explain.