Chapter 3

Write each expression as a sum or difference of logarithms. Example: \(\log \left(m^{2} n^{5}\right)=2 \log m+5 \log n\) $$\log _{b}(\sqrt{r} \sqrt[3]{t})$$

Write each exponential equation in its equivalent logarithmic form. $$100,000=10^{5}$$

Use the following formula for Newton’s Law of Cooling: If you take a hot dinner out of the oven and place it on the kitchen countertop, the dinner cools until it reaches the temperature of the kitchen. Likewise, a glass of ice set on a table in a room eventually melts into a glass of water at that room temperature. The rate at which the hot dinner cools or the ice in the glass melts at any given time is proportional to the difference between its temperature and the temperature of its surroundings (in this case, the room). This is called Newton's law of cooling (or warming) and is modeled by $$T=T_{S}+\left(T_{0}-T_{S}\right) e^{-k t}$$ where \(T\) is the temperature of an object at time \(t, T_{s}\) is the temperature of the surrounding medium, \(T_{0}\) is the temperature of the object at time \(t=0, t\) is the time, and \(k\) is a constant. An apple pie is taken out of the oven with an internal temperature of \(325^{\circ} \mathrm{F}\). It is placed on a rack in a room with a temperature of \(72^{\circ} \mathrm{F}\). After 10 minutes, the temperature of the pie is \(200^{\circ} \mathrm{F}\). What will the temperature of the pie be 30 minutes after coming out of the oven?

Solve the exponential equations. Make sure to isolate the base to a power first. Round our answers to three decimal places. $$3 e^{2 x}=18$$

Write each expression as a sum or difference of logarithms. Example: \(\log \left(m^{2} n^{5}\right)=2 \log m+5 \log n\) $$\log _{b}\left(\frac{r^{1 / 3}}{s^{1 / 2}}\right)$$

Use the following formula for Newton’s Law of Cooling: If you take a hot dinner out of the oven and place it on the kitchen countertop, the dinner cools until it reaches the temperature of the kitchen. Likewise, a glass of ice set on a table in a room eventually melts into a glass of water at that room temperature. The rate at which the hot dinner cools or the ice in the glass melts at any given time is proportional to the difference between its temperature and the temperature of its surroundings (in this case, the room). This is called Newton's law of cooling (or warming) and is modeled by $$T=T_{S}+\left(T_{0}-T_{S}\right) e^{-k t}$$ where \(T\) is the temperature of an object at time \(t, T_{s}\) is the temperature of the surrounding medium, \(T_{0}\) is the temperature of the object at time \(t=0, t\) is the time, and \(k\) is a constant. A cold drink is taken out of an ice chest with a temperature of \(38^{\circ} \mathrm{F}\) and placed on a picnic table with a surrounding temperature of \(75^{\circ} \mathrm{F}\). After 5 minutes, the temperature of the drink is \(45^{\circ} \mathrm{F}\). What will the temperature of the drink be 20 minutes after it is taken out of the chest?

Solve the exponential equations. Make sure to isolate the base to a power first. Round our answers to three decimal places. $$4\left(10^{3 x}\right)=20$$

Write each expression as a sum or difference of logarithms. Example: \(\log \left(m^{2} n^{5}\right)=2 \log m+5 \log n\) $$\log _{b}\left(\frac{r^{4}}{s^{2}}\right)$$

Write each exponential equation in its equivalent logarithmic form. $$7=\sqrt[3]{343}$$

Use the following formula for Newton’s Law of Cooling: If you take a hot dinner out of the oven and place it on the kitchen countertop, the dinner cools until it reaches the temperature of the kitchen. Likewise, a glass of ice set on a table in a room eventually melts into a glass of water at that room temperature. The rate at which the hot dinner cools or the ice in the glass melts at any given time is proportional to the difference between its temperature and the temperature of its surroundings (in this case, the room). This is called Newton's law of cooling (or warming) and is modeled by $$T=T_{S}+\left(T_{0}-T_{S}\right) e^{-k t}$$ where \(T\) is the temperature of an object at time \(t, T_{s}\) is the temperature of the surrounding medium, \(T_{0}\) is the temperature of the object at time \(t=0, t\) is the time, and \(k\) is a constant. A body is discovered in a hotel room. At 7: 00 A.M. a police detective found the body's temperature to be \(85^{\circ} \mathrm{F}\). At 8: 30 A.M. a medical examiner measures the body's temperature to be \(82^{\circ} \mathrm{F}\). Assuming the room in which the body was found had a constant temperature of \(74^{\circ} \mathrm{F}\), how long has the victim been dead? (Normal body temperature is \(98.6^{\circ} \mathrm{F} .\) ).

Solve the exponential equations. Make sure to isolate the base to a power first. Round our answers to three decimal places. $$4 e^{2 x+1}=17$$

Graph the exponential function using transformations. State the \(y\) -intercept, two additional points, the domain, the range, and the horizontal asymptote. $$f(x)=6^{x}$$

Write each expression as a sum or difference of logarithms. Example: \(\log \left(m^{2} n^{5}\right)=2 \log m+5 \log n\) $$\log _{b}\left(\frac{x}{y z}\right)$$

Write each exponential equation in its equivalent logarithmic form. $$\frac{8}{125}=\left(\frac{2}{5}\right)^{3}$$

Use the following formula for Newton’s Law of Cooling: If you take a hot dinner out of the oven and place it on the kitchen countertop, the dinner cools until it reaches the temperature of the kitchen. Likewise, a glass of ice set on a table in a room eventually melts into a glass of water at that room temperature. The rate at which the hot dinner cools or the ice in the glass melts at any given time is proportional to the difference between its temperature and the temperature of its surroundings (in this case, the room). This is called Newton's law of cooling (or warming) and is modeled by $$T=T_{S}+\left(T_{0}-T_{S}\right) e^{-k t}$$ where \(T\) is the temperature of an object at time \(t, T_{s}\) is the temperature of the surrounding medium, \(T_{0}\) is the temperature of the object at time \(t=0, t\) is the time, and \(k\) is a constant. At 4 A.M. a body is found in a park. The police measure the body's temperature to be \(90^{\circ} \mathrm{F} .\) At 5 A.M. the medical examiner arrives and determines the temperature to be \(86^{\circ} \mathrm{F}\). Assuming the temperature of the park was constant at \(60^{\circ} \mathrm{F}\), how long has the victim been dead?

Solve the exponential equations. Make sure to isolate the base to a power first. Round our answers to three decimal places. $$5\left(10^{x^{2}+2 x+1}\right)=13$$

Graph the exponential function using transformations. State the \(y\) -intercept, two additional points, the domain, the range, and the horizontal asymptote. $$f(x)=7^{x}$$

Write each expression as a sum or difference of logarithms. Example: \(\log \left(m^{2} n^{5}\right)=2 \log m+5 \log n\) $$\log _{b}\left(\frac{x y}{z}\right)$$

Write each exponential equation in its equivalent logarithmic form. $$\frac{8}{27}=\left(\frac{2}{3}\right)^{3}$$

Solve the exponential equations. Make sure to isolate the base to a power first. Round our answers to three decimal places. $$3\left(4^{x^{2}-4}\right)=16$$

Write each expression as a sum or difference of logarithms. Example: \(\log \left(m^{2} n^{5}\right)=2 \log m+5 \log n\) $$\log \left(x^{2} \sqrt{x+5}\right)$$

Write each exponential equation in its equivalent logarithmic form. $$3=\left(\frac{1}{27}\right)^{-1 / 3}$$

A new Hyundai Triburon has a book value of \(\$ 22,000\), and after 2 years a book value of \(\$ 14,000 .\) What is the car's value in 4 years? Apply the formula \(N=N_{0} e^{-n t},\) where \(N\) represents the value of the car. Round to the nearest hundred.

Solve the exponential equations. Make sure to isolate the base to a power first. Round our answers to three decimal places. $$7 \cdot\left(\frac{1}{4}\right)^{6-5 x}=3$$

Write each expression as a sum or difference of logarithms. Example: \(\log \left(m^{2} n^{5}\right)=2 \log m+5 \log n\) $$\log [(x-3)(x+2)]$$

Graph the exponential function using transformations. State the \(y\) -intercept, two additional points, the domain, the range, and the horizontal asymptote. $$f(x)=4^{-x}$$

Write each exponential equation in its equivalent logarithmic form. $$4=\left(\frac{1}{1024}\right)^{-1 / 5}$$

A new model BMW convertible coupe is designed and produced in time to appear in North America in the fall. BMW Corporation has a limited number of new models available. The number of new model BMW convertible coupes purchased in North America is given by \(N=\frac{100,000}{1+10 e^{-2 t}},\) where \(t\) is the number of weeks after the BMW is released. a. How many new model BMW convertible coupes will have been purchased 2 weeks after the new model becomes available? b. How many after 30 weeks? c. What is the maximum number of new model BMW convertible coupes that will be sold in North America?

Solve the exponential equations. Make sure to isolate the base to a power first. Round our answers to three decimal places. $$e^{2 x}+7 e^{x}-3=0$$

Write each expression as a sum or difference of logarithms. Example: \(\log \left(m^{2} n^{5}\right)=2 \log m+5 \log n\) $$\ln \left[\frac{x^{3}(x-2)^{2}}{\sqrt{x^{2}+5}}\right]$$

Graph the exponential function using transformations. State the \(y\) -intercept, two additional points, the domain, the range, and the horizontal asymptote. $$f(x)=e^{x}$$

Write each exponential equation in its equivalent logarithmic form. $$e^{x}=6$$

The number of iPhones purchased is given by \(N=\frac{2,000,000}{1+2 e^{-4 t}},\) where \(t\) is the time in weeks after they are made available for purchase. a. How many iPhones are purchased within the first 2 weeks? b. How many iPhones are purchased within the first month?

Solve the exponential equations. Make sure to isolate the base to a power first. Round our answers to three decimal places. $$e^{2 x}-4 e^{x}-5=0$$

Write each expression as a sum or difference of logarithms. Example: \(\log \left(m^{2} n^{5}\right)=2 \log m+5 \log n\) $$\ln \left[\frac{\sqrt{x+3} \sqrt[3]{x-4}}{(x+1)^{4}}\right]$$

Write each exponential equation in its equivalent logarithmic form. $$e^{-x}=4$$

Solve the exponential equations. Make sure to isolate the base to a power first. Round our answers to three decimal places. $$\left(3^{x}-3^{-x}\right)^{2}=0$$

Write each expression as a sum or difference of logarithms. Example: \(\log \left(m^{2} n^{5}\right)=2 \log m+5 \log n\) $$\log \left(\frac{x^{2}-2 x+1}{x^{2}-9}\right)$$

Graph the exponential function using transformations. State the \(y\) -intercept, two additional points, the domain, the range, and the horizontal asymptote. $$f(x)=e^{-x}$$

Write each exponential equation in its equivalent logarithmic form. $$x=y^{z}$$

Solve the exponential equations. Make sure to isolate the base to a power first. Round our answers to three decimal places. $$\left(3^{x}-3^{-x}\right)\left(3^{x}+3^{-x}\right)=0$$

Write each expression as a sum or difference of logarithms. Example: \(\log \left(m^{2} n^{5}\right)=2 \log m+5 \log n\) $$\log \left(\frac{x^{2}-x-2}{x^{2}+3 x-4}\right)$$

Graph the exponential function using transformations. State the \(y\) -intercept, two additional points, the domain, the range, and the horizontal asymptote. $$f(x)=-e^{x}$$

Write each exponential equation in its equivalent logarithmic form. $$z=y^{x}$$

The Virginia Department of Fish and Game stock a mountain lake with 500 trout. Officials believe the lake can support no more than 10,000 trout. The number of trout is given by \(N=\frac{10,000}{1+19 e^{-1.56 t}}\), where \(t\) is time in years. How many years will it take for the trout population to reach \(5000 ?\)

Solve the exponential equations. Make sure to isolate the base to a power first. Round our answers to three decimal places. $$\frac{2}{e^{x}-5}=1$$

Write each expression as a sum or difference of logarithms. Example: \(\log \left(m^{2} n^{5}\right)=2 \log m+5 \log n\) $$\ln \sqrt{\frac{x^{2}+3 x-10}{x^{2}-3 x+2}}$$

Graph the exponential function using transformations. State the \(y\) -intercept, two additional points, the domain, the range, and the horizontal asymptote. $$f(x)=2^{x}-1$$

Evaluate the logarithms exactly (if possible). $$\log _{2} 1$$

The World Wildlife Fund has placed 1000 rare pygmy elephants in a conservation area in Borneo. They believe 1600 pygmy elephants can be supported in this environment. The number of elephants is given by \(N=\frac{1600}{1+0.6 e^{-0.14 t}},\) where \(t\) is time in years. How many years will it take the herd to reach 1200 elephants?