Newtons Law of Cooling is (A) an exponential model. (B) a

In Exercises 1-4, prove that each function in the given pair is the inverse of the other. \(f(x)=e^{2 x}\) and \(g(x)=\ln \left(x^{1 / 2}\right)\)

In Exercises 1-4, prove that each function in the given pair is the inverse of the other. \(f(x)=10^{x / 2}\) and \(g(x)=\log x^2, x>0\)

In Exercises 1-4, prove that each function in the given pair is the inverse of the other. \(f(x)=(1 / 3) \ln x\) and \(g(x)=e^{3 x}\)

In Exercises 1-4, prove that each function in the given pair is the inverse of the other. \(f(x)=3 \log x^2, x>0\) and \(g(x)=10^{x / 6}\)

In Exercises 5 and 6, write the number in scientific notation. The mean distance from Jupiter to the Sun is about 778,300,000 km.

In Exercises 5 and 6, write the number in scientific notation. An atomic nucleus has a diameter of about 0.000000000000001 m.

In Exercises 7 and 8, write the number in decimal form. Avogadro’s number is about \(6.02 \times 10^{23}\) .

In Exercises 7 and 8, write the number in decimal form. The atomic mass unit is about \(1.66 \times 10^{-27} \ \mathrm {kg}\) .

In Exercises 9 and 10, use scientific notation to simplify the expression; leave your answer in scientific notation. (186,000) (31,000,000)

In Exercises 9 and 10, use scientific notation to simplify the expression; leave your answer in scientific notation. \(\frac {0.0000008}{0.000005}\)

In Exercises 1-10, find the exact solution algebraically, and check it by substituting into the original equation. \(36\left(\frac{1}{3}\right)^{x / 5}=4\)

In Exercises 1-10, find the exact solution algebraically, and check it by substituting into the original equation. \(32\left(\frac{1}{4}\right)^{x / 3}=2\)

In Exercises 1-10, find the exact solution algebraically, and check it by substituting into the original equation. \(2 \cdot 5^{x / 4}=250\)

In Exercises 1-10, find the exact solution algebraically, and check it by substituting into the original equation. \(3 \cdot 4^{x / 2}=96\)

In Exercises 1-10, find the exact solution algebraically, and check it by substituting into the original equation. \(2\left(10^{-x / 3}\right)=20\)

In Exercises 1-10, find the exact solution algebraically, and check it by substituting into the original equation. \(3\left(5^{-x / 4}\right)=15\)

In Exercises 1-10, find the exact solution algebraically, and check it by substituting into the original equation. log x=4

In Exercises 1-10, find the exact solution algebraically, and check it by substituting into the original equation. \(\log _2 x=5\)

In Exercises 1-10, find the exact solution algebraically, and check it by substituting into the original equation. \(\log _4(x-5)=-1\)

In Exercises 1-10, find the exact solution algebraically, and check it by substituting into the original equation. \(\log _4(1-x)=1\)

In Exercises 11-18, solve each equation algebraically. Obtain a numerical approximation for your solution and check it by substituting into the original equation. \(1.06^x=4.1\)

In Exercises 11-18, solve each equation algebraically. Obtain a numerical approximation for your solution and check it by substituting into the original equation. \(0.98^x=1.6\)

In Exercises 11-18, solve each equation algebraically. Obtain a numerical approximation for your solution and check it by substituting into the original equation. \(50 e^{0.035 x}=200\)

In Exercises 11-18, solve each equation algebraically. Obtain a numerical approximation for your solution and check it by substituting into the original equation. \(80 e^{0.045 x}=240\)

In Exercises 11-18, solve each equation algebraically. Obtain a numerical approximation for your solution and check it by substituting into the original equation. \(3+2 e^{-x}=6\)

In Exercises 11-18, solve each equation algebraically. Obtain a numerical approximation for your solution and check it by substituting into the original equation. \(7-3 e^{-x}=2\)

In Exercises 11-18, solve each equation algebraically. Obtain a numerical approximation for your solution and check it by substituting into the original equation. 3 ln (x - 3) + 4 = 5

In Exercises 11-18, solve each equation algebraically. Obtain a numerical approximation for your solution and check it by substituting into the original equation. 3 - log (x + 2) = 5

In Exercises 19-24, state the domain of each function. Then match the function with its graph. (Each graph shown has a window of [-4.7,4.7] by [-3.1,3.1]). f(x) = log [x(x + 1)]

In Exercises 19-24, state the domain of each function. Then match the function with its graph. (Each graph shown has a window of [-4.7,4.7] by [-3.1,3.1]). g(x) = log x + log (x + 1)

In Exercises 19-24, state the domain of each function. Then match the function with its graph. (Each graph shown has a window of [-4.7,4.7] by [-3.1,3.1]). \(f(x)=\ln \frac{x}{x+1}\)

In Exercises 19-24, state the domain of each function. Then match the function with its graph. (Each graph shown has a window of [-4.7,4.7] by [-3.1,3.1]). g(x) = ln x - ln (x + 1)

In Exercises 19-24, state the domain of each function. Then match the function with its graph. (Each graph shown has a window of [-4.7,4.7] by [-3.1,3.1]). f(x) = 2 ln x

In Exercises 19-24, state the domain of each function. Then match the function with its graph. (Each graph shown has a window of [-4.7,4.7] by [-3.1,3.1]). \(g(x)=\ln x^2\)

In Exercises 25–38, solve each equation by the method of your choice. Support your solution by a second method. \(\log x^2=6\)

In Exercises 25–38, solve each equation by the method of your choice. Support your solution by a second method. \( \ln x^2=4\)

In Exercises 25–38, solve each equation by the method of your choice. Support your solution by a second method. \(\log x^4=2\)

In Exercises 25–38, solve each equation by the method of your choice. Support your solution by a second method. \(\ln x^6=12\)

In Exercises 25–38, solve each equation by the method of your choice. Support your solution by a second method. \(\frac{2^x-2^{-x}}{3}=4\)

In Exercises 25–38, solve each equation by the method of your choice. Support your solution by a second method. \(\frac{2^x+2^{-x}}{2}=3\)

In Exercises 25–38, solve each equation by the method of your choice. Support your solution by a second method. \(\frac{e^x+e^{-x}}{2}=4\)

In Exercises 25–38, solve each equation by the method of your choice. Support your solution by a second method. \(2 e^{2 x}+5 e^x-3=0\)

In Exercises 25–38, solve each equation by the method of your choice. Support your solution by a second method. \(\frac{500}{1+25 e^{0.3 x}}=200\)

In Exercises 25–38, solve each equation by the method of your choice. Support your solution by a second method. \(\frac{400}{1+95 e^{-0.6 x}}=150\)

In Exercises 25–38, solve each equation by the method of your choice. Support your solution by a second method. \(\frac{1}{2} \ln (x+3)-\ln x=0\)

In Exercises 25–38, solve each equation by the method of your choice. Support your solution by a second method. \(\log x-\frac{1}{2} \log (x+4)=1\)

In Exercises 25–38, solve each equation by the method of your choice. Support your solution by a second method. ln (x - 3)+ ln (x + 4)=3 ln 2

In Exercises 25–38, solve each equation by the method of your choice. Support your solution by a second method. log (x-2) + log (x + 5) = 2 log 3

In Exercises 39-44, determine by how many orders of magnitude the quantities differ. A $100 bill and a dime

In Exercises 39-44, determine by how many orders of magnitude the quantities differ. A canary weighing 20 g and a hen weighing 2 kg

In Exercises 39-44, determine by how many orders of magnitude the quantities differ. An earthquake rated 7 on the Richter scale and one rated 5.5

In Exercises 39-44, determine by how many orders of magnitude the quantities differ. Lemon juice with pH = 2.3 and beer with pH = 4.1

In Exercises 39-44, determine by how many orders of magnitude the quantities differ. The sound intensities of a riveter at 95 dB and ordinary conversation at 65 dB

In Exercises 39-44, determine by how many orders of magnitude the quantities differ. The sound intensities of city traffic at 70 dB and rustling leaves at 10 dB

Comparing Earthquakes How many times more severe was the 1978 Mexico City earthquake (R=7.9) than the 1994 Los Angeles earthquake (R=6.6)?

Comparing Earrthquakes How many times more severe was the 1995 Kobe, Japan, earthquake (R=7.2) than the 1994 Los Angeles earthquake (R=6.6)?

Chemical Acidity The pH of carbonated water is 3.9 and the pH of household ammonia is 11.9. (a) What are their hydrogen-ion concentrations? (b) How many times greater is the hydrogen-ion concentration of carbonated water than that of ammonia? (c) By how many orders of magnitude do the concentrations differ?

Chemical Acidity Stomach acid has a pH of about 2.0, and blood has a pH of 7.4. (a) What are their hydrogen-ion concentrations? (b) How many times greater is the hydrogen-ion concentration of stomach acid than that of blood? (c) By how many orders of magnitude do the concentrations differ?

Newton's Law of Cooling A cup of coffee has cooled from \(92^{\circ} \mathrm{C}\) to \(50^{\circ} \mathrm{C}\) after 12 min in a room at \(22^{\circ} \mathrm{C}\). How long will the cup take to cool to \(30^{\circ} \mathrm{C}\) ?

Newton's Law of Cooling A cake is removed from an oven at \(350^{\circ} \mathrm{F}\) and cools to \(120^{\circ} \mathrm{F}\) after 20 min in a room at \(65^{\circ} \mathrm{F}\). How long will the cake take to cool to \(90^{\circ} \mathrm{F}\)?

Newton's Law of Cooling Experiment A thermometer is removed from a cup of coffee and placed in water with a temperature \(\left(T_m\right)\) of \(10^{\circ} \mathrm{C}\). The data in Table 3.24 were collected over the next 30 sec. (a) Draw a scatter plot of the data \(T-T_m\). (b) Find an exponential regression equation for the \(T-T_m\) data. Superimpose its graph on the scatter plot. (c) Estimate the thermometer reading when it was removed from the coffee.

Newton’s Law of Cooling Experiment A thermometer was removed from a cup of hot chocolate and placed in a saline solution with temperature \(T_m=0^\circ \mathrm C\) . The data in Table 3.25 were collected over the next 30 sec. (a) Draw a scatter plot of the data \(T-T_m\) . (b) Find an exponential regression equation for the \(T-T_m\) data. Superimpose its graph on the scatter plot. (c) Estimate the thermometer reading when it was removed from the hot chocolate.

Penicillin Use The use of penicillin became so widespread in the 1980s in Hungary that it became practically useless against common sinus and ear infections. Now the use of more effective antibiotics has caused a decline in penicillin resistance. The bar graph shows the use of penicillin in Hungary for selected years. (a) From the bar graph we read the data pairs to be approximately )1,11), (8,6), (15,4.8), (16,4), and (17,2.5), using t = 1 for 1976, t = 8 for 1983, and so on. Complete a scatter plot for these data. (b) Writing to Learn Discuss whether the bar graph shown or the scatter plot that you completed best represents the data and why.

Writing to Learn Which regression model would you use for the data in Exercise 53? Discuss various options, and explain why you chose the model you did. Support your writing with tables and graphs as needed.

Writing to Learn In Exercises 55–58, tables of (x, y) data pairs are given. Determine whether a linear, logarithmic, exponential, or power regression equation is the best model for the data. Explain your choice. Support your writing with tables and graphs as needed. \(\begin{array}{c|cccc} x & 1 & 2 & 3 & 4 \\ \hline y & 3 & 4.4 & 5.2 & 5.8 \end{array}\)

Writing to Learn In Exercises 55–58, tables of (x, y) data pairs are given. Determine whether a linear, logarithmic, exponential, or power regression equation is the best model for the data. Explain your choice. Support your writing with tables and graphs as needed. \(\begin{array}{c|cccc} x & 1 & 2 & 3 & 4 \\ \hline y & 6 & 18 & 54 & 162 \end{array}\)

Writing to Learn In Exercises 55–58, tables of (x, y) data pairs are given. Determine whether a linear, logarithmic, exponential, or power regression equation is the best model for the data. Explain your choice. Support your writing with tables and graphs as needed. \(\begin{array}{c|cccc} x & 1 & 2 & 3 & 4 \\ \hline y & 3 & 6 & 12 & 24 \end{array}\)

Writing to Learn In Exercises 55–58, tables of (x, y) data pairs are given. Determine whether a linear, logarithmic, exponential, or power regression equation is the best model for the data. Explain your choice. Support your writing with tables and graphs as needed. \(\begin{array}{c|cccc} x & 1 & 2 & 3 & 4 \\ \hline y & 5 & 7 & 9 & 11 \end{array}\)

True or False The order of magnitude of a positive number is its natural logarithm. Justify your answer.

True or False According to Newton’s Law of Cooling, an object will approach the temperature of the medium that surrounds it. Justify your answer.

In Exercises 61–64, solve the problem without using a calculator. Multiple Choice Solve \(2^{3 x-1}=32\) (A) x = 1 (B) x = 2 (C) x = 4 (D) x = 11 (E) x = 13

In Exercises 61–64, solve the problem without using a calculator. Multiple Choice Solve \(\ln x=-1\) (A) x = -1 (B) x = 1/e (C) x = 1 (D) x = e (E) No solution is possible.

In Exercises 61–64, solve the problem without using a calculator. Multiple Choice How many times more severe was the 2001 earthquake in Arequipa, Peru \(\left(R_{1}=8.1\right)\), than the 1998 double earthquake in Takhar province, Afghanistan \(\left(R_{2}=6.1\right)\)? (A) 2 (B) 6.1 (C) x = 1 (D) 14.2 (E) 100

In Exercises 61–64, solve the problem without using a calculator. Multiple Choice Newton’s Law of Cooling is (A) an exponential model. (B) a linear model. (C) a logarithmic model. (D) a logistic model. (E) a power model.

In Exercises 65 and 66, use the data in Table 3.26. Determine whether a linear, logarithmic, exponential, power, or logistic regression equation is the best model for the data. Explain your choice. Support your writing with tables and graphs as needed. Writing to Learn Modeling Population Which regression equation is the best model for Alaska’s population?

In Exercises 65 and 66, use the data in Table 3.26. Determine whether a linear, logarithmic, exponential, power, or logistic regression equation is the best model for the data. Explain your choice. Support your writing with tables and graphs as needed. Writing to Learn Modeling Population Which regression equation is the best model for Hawaii’s population?

Group Activity Normal Distribution The function \(f(x)=k \cdot e^{-c x^{2}}\), where c and k are positive constants, is a bell-shaped curve that is useful in probability and statistics. (a) Graph f for c = 1 and k = 0.1 , 0.5, 1, 2, 10. Explain the effect of changing k. (b) Graph f for k = 1 and c = 0.1, 0.5, 1, 2, 10. Explain the effect of changing c.

Writing to Learn Prove if \(u / v=10^{n}\) for \(u>0\) and \(v>0\), then log u - log v = n. Explain how this result relates to powers of ten and orders of magnitude.

Potential Energy The potential energy E (the energy stored for use at a later time) between two ions in a certain molecular structure is modeled by the function \(E=-\frac{5.6}{r}+10 e^{-r / 3}\) where r is the distance separating the nuclei. (a) Writing to Learn Graph this function in the window [-10,10] by [-10,30], and explain which portion of the graph does not represent this potential energy situation. (b) Identify a viewing window that shows that portion of the graph (with \(r \leq 10\)) which represents this situation, and find the maximum value for E.

In Example 8, the Newton’s Law of Cooling model was \(T(t)-T_{m}=\left(T_{0}-T_{m}\right) e^{-k t}=61.656 \times 0.92770^{t}\) Determine the value of k.

Justify the conclusion made about natural logarithmic regression on page 299.

Justify the conclusion made about power regression on page 299.

In Exercises 73–78, solve the equation or inequality. \(e^{x}+x=5\)

In Exercises 73–78, solve the equation or inequality. \(e^{2 x}-8 x+1=0\)

In Exercises 73–78, solve the equation or inequality. \(e^{x}<5+\ln x\)

In Exercises 73–78, solve the equation or inequality. \(\ln |x|-e^{2 x} \geq 3\)

In Exercises 73–78, solve the equation or inequality. \(2 \log x-4 \log 3>0\)

In Exercises 73–78, solve the equation or inequality. \(2 \log (x+1)-2 \log 6<0\)