Chapter 4

Refer to the following. The magnitude of an earthquake is measured on the Richter scale using the formula $$R(I)=\log \left(\frac{I}{I_{0}}\right)$$ where I represents the actual intensity of the earthquake and \(I_{0}\) is a baseline intensity used for comparison. Richter Scale If the intensity of an earthquake is 10,000 times the baseline intensity \(I_{0},\) what is its magnitude on the Richter scale?

Give an example of a function that is its own inverse.

A new car that costs $$\$ 25,000$$ depreciates to $$80 \%$$ of its value in 3 years. (a) Assume the depreciation is linear. What is the linear function that models the value of this car \(t\) years after purchase? (b) Assume the value of the car is given by an exponential function \(y=A e^{h t},\) where \(A\) is the initial price of the car. Find the value of the constant \(k\) and the exponential function. (c) Using the linear model found in part (a), find the value of the car 5 years after purchase. Do the same using the exponential model found in part (b). (d) Graph both models using a graphing utility. Which model do you think is more realistic, and why?

Explain why the function \(f(x)=2^{x}\) has no vertical asymptotes (review Section 4.6).

Refer to the following. The magnitude of an earthquake is measured on the Richter scale using the formula $$R(I)=\log \left(\frac{I}{I_{0}}\right)$$ where I represents the actual intensity of the earthquake and \(I_{0}\) is a baseline intensity used for comparison. Richter Scale If the intensity of an earthquake is a million times the baseline intensity \(I_{0},\) what is its magnitude on the Richter scale?

The function \(f(x)=x^{6}\) is not one-to-one. How can the domain of \(f\) be restricted to produce a one-to-one function?

Pesticides decay at different rates depending on the pH level of the water contained in the pesticide solution. The pH scale measures the acidity of a solution. The lower the pH value, the more acidic the solution. When produced with water that has a pH of 6.0, the pesticide chemical known as malathion has a half-life of 8 days; that is, half the initial amount of malathion will remain after 8 days. However, if it is produced with water that has a pH of \(7.0,\) the half-life of malathion decreases to 3 days. (Source: Cooperative Extension Program, University of Missouri) (a) Assume the initial amount of malathion is 5 milligrams. Find an exponential function of the form \(A(t)=A_{0} e^{k t}\) that gives the amount of malathion that remains after \(t\) days if it is produced with water that has a pH of 6.0 (b) Assume the initial amount of malathion is 5 milligrams. Find an exponential function of the form \(B(t)=B_{0} e^{t t}\) that gives the amount of malathion that remains after \(t\) days if it is produced with water that has a pH of 7.0 (c) How long will it take for the amount of malathion in each of the solutions in parts (a) and (b) to decay to 3 milligrams? (d) If the malathion is to be stored for a few days before use, which of the two solutions would be more effective, and why? 4 (e) Graph the two exponential functions in the same viewing window and describe how the graphs illustrate the differing decay rates.

The function \(f(x)=|x+2|\) is not one-to-one. How can the domain of \(f\) be restricted to produce a one-to-one function?

Refer to the following. The magnitude of an earthquake is measured on the Richter scale using the formula $$R(I)=\log \left(\frac{I}{I_{0}}\right)$$ where I represents the actual intensity of the earthquake and \(I_{0}\) is a baseline intensity used for comparison. Earthquake Intensity What is the ratio of the intensity of a quake that measures 7.1 on the Richter scale to the intensity of one that measures \(4.2 ?\)

Exercises 88 and 89 refer to the following. The pH of a chemical solution is given by \(\mathrm{pH}=-\log \left[\mathrm{H}^{+}\right],\) where \(\left[\mathrm{H}^{+}\right]\) is the concentration of hydrogen ions in the solution, in units of moles per liter. (One mole is \(6.02 \times 10^{23}\) molecules.) Chemistry Find the \(\mathrm{pH}\) of a solution for which \(\left[\mathrm{H}^{+}\right]=0.001\) mole per liter.

Exercises 88 and 89 refer to the following. The pH of a chemical solution is given by \(\mathrm{pH}=-\log \left[\mathrm{H}^{+}\right],\) where \(\left[\mathrm{H}^{+}\right]\) is the concentration of hydrogen ions in the solution, in units of moles per liter. (One mole is \(6.02 \times 10^{23}\) molecules.) Chemistry Find the \(\mathrm{pH}\) of a solution for which \(\left[\mathrm{H}^{+}\right]=10^{-4}\) mole per liter.

This set of exercises will draw on the ideas presented in this section and your general math background. Do the equations \(\ln x^{2}=1\) and \(2 \ln x=1\) have the same solutions? Explain.

Astronomy The brightness of a star is designated on a numerical scale called magnitude, which is defined by the formula $$ M(I)=-\log _{2.5} \frac{I}{I_{0}} $$ where \(I\) is the energy intensity of the star and \(I_{0}\) is the baseline intensity used for comparison. A decrease of 1 unit in magnitude represents an increase in energy intensity of a factor of \(2.5 .\) (Source: National Acronautics and Space Agency) (a) If the star Spica has magnitude \(1,\) find its intensity in terms of \(I_{0}\) (b) The star Sirius, the brightest star other than the sun, has magnitude \(-1.46 .\) Find its intensity in terms of In. What is the ratio of the intensity of Sirius to that of Spica?

This set of exercises will draw on the ideas presented in this section and your general math background. Explain why the equation \(2 e^{x}=-1\) has no solution.

Computer Science Computer programs perform many kinds of sorting. It is preferable to use the least amount of computer time to do the sorting, where the measure of computer time is the number of operations the computer needs to perform. Two methods of sorting are the bubble sort and the heap sort. It is known that the bubble sort algorithm requires approximately \(n^{2}\) operations to sort a list of \(n\) items, while the heap sort algorithm requires approximately \(n\) log \(n\) operations to sort \(n\) items. (a) To sort 100 items, how many operations are required by the bubble sort? by the heap sort? (b) Make a table listing the number of operations required for the bubble sort to sort a list of \(n\) items, with \(n\) ranging from 5 to 20 , in steps of \(5 .\) If the number of items sorted is doubled from 10 to \(20,\) what is the corresponding increase in the number of operations? (c) Rework part (b) for the heap sort. (d) Which algorithm, the bubble sort or the heap sort, is more efficient? Why? 4 (e) \(=\) In the same viewing window, graph the functions that give the number of operations for the bubble sort and for the heap sort. Let \(n\) range from 1 to 20 . Which function is growing faster, and why? Note that you will have to choose the vertical scale carefully so that the \(n\) log \(n\) function does not get "squashed."

This set of exercises will draw on the ideas presented in this section and your general math background. What is wrong with the following step? $$\log x+\log (x+1)=0 \Rightarrow x(x+1)=0$$

Ecology The pH scale measures the level of acidity of a solution on a logarithmic scale. A pH of 7.0 is considered neutral. If the \(p H\) is less than \(7.0,\) then the solution is acidic. The lower the \(\mathrm{pH}\), the more acidic the solution. since the pH scale is logarithmic, a single unit decrease in pH represents a tenfold increase in the acidity level. (a) The average pH of rainfall in the northeastern part of the United States is \(4.5 .\) Normal rainfall has a pH of \(5.5 .\) Compared to normal rainfall, how many times more acidic is the rainfall in the northeastern United States, on average? Explain. (Source: U.S. Environmental Protection Agency) (b) Because of increases in the acidity of rain, many lakes in the northeastern United States have become more acidic. The degree to which acidity can be tolerated by fish in these lakes depends on the species. The yellow perch can easily tolerate a pH of 4.0, while the common shiner cannot easily tolerate pH levels below \(6.0 .\) Which species is more likely to survive in a more acidic environment, and why? What is the ratio of the acidity levels that are easily tolerated by the yellow perch and the common shiner? Explain. (Source: U.S. Environmental Protection Agency)

This set of exercises will draw on the ideas presented in this section and your general math background. What is wrong with the following step? $$2^{x+5}=3^{4 x} \Rightarrow x+5=4 x$$

Solve using any method, and eliminate extraneous solutions. $$\ln (\log x)=1$$

This set of exercises will draw on the ideas presented in this section and your general math background. Explain why In 4 is between 1 and \(2,\) without using a calculator.

Solve using any method, and eliminate extraneous solutions. $$e^{\log x}=e$$

Explain how you could use the following table of values for the function \(f(x)=10^{x}\) to find the given quantity. $$\begin{array}{cc} x & f(x)=10^{x} \\ 0.4771 & 3 \\ 0.5 & \sqrt{10} \\ 3 & 1000 \\ -0.3010 & 0.5 \\ \sqrt{10} & 1452 \end{array}$$ $$. \log 1000$$

Solve using any method, and eliminate extraneous solutions. $$\log _{5}|x-2|=2$$

Explain how you could use the following table of values for the function \(f(x)=10^{x}\) to find the given quantity. $$\begin{array}{cc} x & f(x)=10^{x} \\ 0.4771 & 3 \\ 0.5 & \sqrt{10} \\ 3 & 1000 \\ -0.3010 & 0.5 \\ \sqrt{10} & 1452 \end{array}$$ $$\log 3$$

Solve using any method, and eliminate extraneous solutions. $$\ln |2 x-3|=1$$

Explain how you could use the following table of values for the function \(f(x)=10^{x}\) to find the given quantity. $$\begin{array}{cc} x & f(x)=10^{x} \\ 0.4771 & 3 \\ 0.5 & \sqrt{10} \\ 3 & 1000 \\ -0.3010 & 0.5 \\ \sqrt{10} & 1452 \end{array}$$ $$\log 0.5$$

Explain how you could use the following table of values for the function \(f(x)=10^{x}\) to find the given quantity. $$\begin{array}{cc} x & f(x)=10^{x} \\ 0.4771 & 3 \\ 0.5 & \sqrt{10} \\ 3 & 1000 \\ -0.3010 & 0.5 \\ \sqrt{10} & 1452 \end{array}$$ $$\log \sqrt{10}$$

The graph of \(f(x)=a \log x\) passes through the point \((10,3) .\) Find \(a\) and thus the complete expression for \(f\) Check your answer by graphing \(f\).

The graph of \(f(x)=A \ln x+B\) passes through the points (1,2) and \((e, 4)\) (a) Find \(A\) and \(B\) using the given points. (b) Check your answer by graphing \(f\)

Sketch graphs of the two functions to show that \(\log _{1 / 2} x=-\log _{2} x .\) (The equality can be established algebraically by techniques in the following section.)

Find the domains of \(f(x)=2 \ln x\) and \(g(x)=\ln x^{2}\) Graph these functions in separate viewing windows. Where are the graphs identical? Explain in terms of the domain you found for each function.