Chapter 4

Use your graphing utility to graph each side of the equation in the same viewing rectangle. Then use the \(x\) -coordinate of the intersection point to find the equation's solution set Verify this value by direct substitution into the equation. $$ \log (x-15)+\log x=2 $$

Write as a single term that does not contain a logarithm: \(e^{\ln 8 x^{3}-\ln 2 x^{2}}\)

Use your graphing utility to graph each side of the equation in the same viewing rectangle. Then use the \(x\) -coordinate of the intersection point to find the equation's solution set Verify this value by direct substitution into the equation. $$ 3^{x}=2 x+3 $$

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 elapsed before the average score fell below 65.

Use your graphing utility to graph each side of the equation in the same viewing rectangle. Then use the \(x\) -coordinate of the intersection point to find the equation's solution set Verify this value by direct substitution into the equation. $$ 5^{x}=3 x+4 $$

In parts (a)-(c), 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). Generalize this observation by writing an 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 _____.

Hurricanes are one of nature's most destructive forces. These low-pressure areas often have diameters of over 500 miles. The function \(f(x)=0.48 \ln (x+1)+27\) models the barometric air pressure, \(f(x),\) in inches of mercury, at a distance of \(x\) miles from the eye of a hurricane. Use this function to solve Graph the function in a \([0,500,50]\) by \([27,30,1]\) viewing rectangle. What does the shape of the graph indicate about barometric air pressure as the distance from the eye increases?

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. $$y=x, y=\sqrt{x}, y=e^{x}, y=\ln x, y=x^{x}, y=x^{2}$$

Exercises \(134-136\) will help you prepare for the material covered in the next section. $$ \text { Solve for } x: a(x-2)=b(2 x+3) $$

Hurricanes are one of nature's most destructive forces. These low-pressure areas often have diameters of over 500 miles. The function \(f(x)=0.48 \ln (x+1)+27\) models the barometric air pressure, \(f(x),\) in inches of mercury, at a distance of \(x\) miles from the eye of a hurricane. Use this function to solve Use an equation to answer this question: How far from the eye of a hurricane is the barometric air pressure 29 inches of mercury? Use the \(\mathrm{TRACE}\) and \(\mathrm{ZOOM}\), features or the intersect command of your graphing utility to verify your answer.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I've noticed that exponential functions and logarithmic functions exhibit inverse, or opposite, behavior in many ways. For example, a vertical translation shifts an exponential function's horizontal asymptote and a horizontal translation shifts a logarithmic function's vertical asymptote.

Exercises \(134-136\) will help you prepare for the material covered in the next section. Solve: \(x(x-7)=3\)

The function \(P(t)=145 e^{-0.092 t}\) models a runner's pulse, \(P(t),\) in beats per minute, \(t\) minutes after a race, where \(0 \leq t \leq 15 .\) Graph the function using a graphing utility. TRACE along the graph and determine after how many minutes the runner's pulse will be 70 beats per minute. Round to the nearest tenth of a minute. Verify your observation algebraically.

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 \(8^{2}=64\).

Exercises \(134-136\) will help you prepare for the material covered in the next section. $$\text { Solve: } \frac{x+2}{4 x+3}=\frac{1}{x}$$

The function \(W(t)=2600\left(1-0.51 e^{-0.075 t}\right)^{3}\) models the weight, \(W(t),\) in kilograms, of a female African elephant at age \(t\) years. (1 kilogram \(=2.2\) pounds) Use a graphing utility to graph the function. Then TRACE along the curve to estimate the age of an adult female elephant weighing 1800 kilograms.

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. Because the equations $$ \log (3 x+1)=5 \text { and } \log (3 x+1)=\log 5 $$ are similar, I solved them using the same method.

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$\frac{\log _{2} 8}{\log _{2} 4}=\frac{8}{4}$$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I can solve \(4^{x}=15\) by writing the equation in logarithmic form.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. It's important for me to check that the proposed solution of an equation with logarithms gives only logarithms of positive numbers in the original equation.

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. \(\log _{b} x\) is the exponent to which \(b\) must be raised to obtain \(x\).

Without using a calculator, find the exact value of \(\log _{4}\left[\log _{3}\left(\log _{2} 8\right)\right]\).

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. Examples of exponential equations include \(10^{x}-5.71\) \(e^{x}-0.72,\) and \(x^{10}-5.71\)

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

If \(\$ 4000\) is deposited into an account paying \(3 \%\) interest compounded annually and at the same time \(\$ 2000\) is deposited into an account paying \(5 \%\) interest compounded annually, after how long will the two accounts have the same balance? Round to the nearest year.

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 \(x=5,6\) \(7, \ldots, 15,\) 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.

Solve each equation. Check each proposed solution by direct substitution or with a graphing utility. $$ (\ln x)^{2}=\ln x^{2} $$

Will help you prepare for the material covered in the next section. In each exercise, evaluate the indicated logarithmic expressions without using a calculator. a. Evaluate: \(\log _{2} 32\) b. Evaluate: \(\log _{2} 8+\log _{2} 4\) c. What can you conclude about \(\log _{2} 32,\) or \(\log _{2}(8 \cdot 4) ?\)

Solve each equation. Check each proposed solution by direct substitution or with a graphing utility. $$ (\log x)(2 \log x+1)=6 $$

Will help you prepare for the material covered in the next section. In each exercise, evaluate the indicated logarithmic expressions without using a calculator. a. Evaluate: \(\log _{2} 16\) b. Evaluate: \(\log _{2} 32-\log _{2} 2\) c. What can you conclude about $$\log _{2} 16, \text { or } \log _{2}\left(\frac{32}{2}\right) ?$$

Solve each equation. Check each proposed solution by direct substitution or with a graphing utility. $$ \ln (\ln x)=0 $$

Will help you prepare for the material covered in the next section. In each exercise, evaluate the indicated logarithmic expressions without using a calculator. a. Evaluate: \(\log _{3} 81\) b. Evaluate: \(2 \log _{3} 9\) c. What can you conclude about $$\log _{3} 81, \text { or } \log _{3} 9^{2} ?$$

Research applications of logarithmic functions as mathematical models and plan a seminar based on your group's research. Each group member should research one of the following areas or any other area of interest: \(\mathrm{pH}\) (acidity of solutions), intensity of sound (decibels), brightness of stars, human memory, progress over time in a sport, profit over time. For the area that you select, explain how logarithmic functions are used and provide examples.

Exercises \(150-152\) will help you prepare for the material covered in the next section. The formula \(A=10 e^{-0.003 t}\) models the population of Hungary, \(A,\) in millions, \(t\) years after 2006 a. Find Hungary's population, in millions, for \(2006,2007\), \(2008,\) and \(2009 .\) Round to two decimal places b. Is Hungary's population increasing or decreasing?

Exercises \(150-152\) will help you prepare for the material covered in the next section. a. Simplify: \(e^{\ln 3}\) b. Use your simplification from part (a) to rewrite \(3^{x}\) in terms of base \(e\)

Exercises \(150-152\) will help you prepare for the material covered in the next section. U.S. soldiers fight Russian troops who have invaded New York City. Incoming missiles from Russian submarines and warships ravage the Manhattan skyline. It's just another scenario for the multi-billion-dollar video games Call of Duty, which have sold more than 100 million games since the franchise's birth in 2003 . The table shows the annual retail sales for Call of Duty video games from 2004 through 2010 . Create a scatter plot for the data. Based on the shape of the scatter plot, would a logarithmic function, an exponential function, or a linear function be the best choice for modeling the data? $$ \begin{array}{cc} \hline \text { Year } & \begin{array}{c} \text { Retail Sales } \\ \text { (millions of dollars) } \end{array} \\ \hline 2004 & 56 \\ 2005 & 101 \\ 2006 & 196 \\ 2007 & 352 \\ 2008 & 436 \\ 2009 & 778 \\ 2010 & 980 \end{array} $$