Chapter 4

Evaluate the expression. (a) \(\log _{2} 32\) (b) \(\log _{8} 8^{17}\) (c) \(\log _{6} 1\)

These exercises use the radioactive decay model. After 3 days a sample of radon-222 has decayed to \(58 \%\) of its original amount. (a) What is the half-life of radon- \(222 ?\) (b) How long will it take the sample to decay to \(20 \%\) of its original amount?

Radioactive Decay Doctors use radioactive iodine as a tracer in diagnosing certain thyroid gland disorders. This type of iodine decays in such a way that the mass remaining after \(t\) days is given by the function $$m(t)=6 e^{-0.087 t}$$ where \(m(t)\) is measured in grams. (a) Find the mass at time \(t=0.\) (b) How much of the mass remains after 20 days?

Use the Laws of Logarithms to expand the expression. $$\log _{5} \frac{x}{2}$$

Find the solution of the exponential equation, rounded to four decimal places. $$2^{3 x+1}=3^{x-2}$$

Evaluate the expression. (a) \(\log _{3}\left(\frac{1}{27}\right)\) (b) \(\log _{10} \sqrt{10}\) (c) \(\log _{5} 0.2\)

These exercises use the radioactive decay model. A wooden artifact from an ancient tomb contains \(65 \%\) of the carbon-14 that is present in living trees. How long ago was the artifact made? (The half-life of carbon- 14 is 5730 years.)

Use the Laws of Logarithms to expand the expression. $$\log 6^{10}$$

Evaluate the expression. (a) \(\log _{5} 125\) (b) \(\log _{49} 7\) (c) \(\log _{9} \sqrt{3}\)

Find the solution of the exponential equation, rounded to four decimal places. $$7^{x / 2}=5^{1-x}$$

These exercises use the radioactive decay model. The burial cloth of an Egyptian mummy is estimated to contain \(59 \%\) of the carbon- 14 it contained originally. How long ago was the mummy buried? (The half-life of carbon- 14 is 5730 years.)

Mixtures and Concentrations A 50 -gallon barrel is filled completely with pure water. Salt water with a concentration of 0.3 Ib/gal is then pumped into the barrel, and the resulting mixture overflows at the same rate. The amount of salt in the barrel at time \(t\) is given by $$Q(t)=15\left(1-e^{-0.04 t}\right)$$ where \(t\) is measured in minutes and \(Q(t)\) is measured in pounds. (a) How much salt is in the barrel after 5 min? (b) How much salt is in the barrel after 10 min? (c) Draw a graph of the function \(Q(t).\) (d) Use the graph in part (c) to determine the value that the amount of salt in the barrel approaches as \(t\) becomes large. Is this what you would expect?

Use the Laws of Logarithms to expand the expression. $$\ln \sqrt{z}$$

Evaluate the expression. (a) \(2^{\log _{2} 37}\) (b) \(3^{\log _{3} 8}\) (c) \(e^{\ln \sqrt{5}}\)

These exercises use Newton’s Law of Cooling. A hot bowl of soup is served at a dinner party. It starts to cool according to Newton's Law of Cooling, so its temperature at time \(t\) is given by $$T(t)=65+145 e^{-0.05 t}$$ where \(t\) is measured in minutes and \(T\) is measured in "F. (a) What is the initial temperature of the soup? (b) What is the temperature after 10 min? (c) After how long will the temperature be \(100^{\circ} \mathrm{F} ?\)

Find the solution of the exponential equation, rounded to four decimal places. $$\frac{50}{1+e^{-x}}=4$$

Use the Laws of Logarithms to expand the expression. $$\log _{2}\left(A B^{2}\right)$$

Evaluate the expression. (a) \(e^{\ln \pi}\) (b) \(10^{\log 5}\) (c) \(10^{\log 87}\)

These exercises use Newton’s Law of Cooling. Newton's Law of Cooling is used in homicide investigations to determine the time of death. The normal body temperature is \(98.6^{\circ} \mathrm{F}\). Immediately following death, the body begins to cool. It has been determined experimentally that the constant in Newton's Law of Cooling is approximately \(k=0.1947,\) assuming that time is measured in hours. Suppose that the temperature of the surroundings is \(60^{\circ} \mathrm{F}\). (a) Find a function \(T(t)\) that models the temperature \(t\) hours after death. (b) If the temperature of the body is now \(72^{\circ} \mathrm{F}\), how long ago was the time of death?

Find the solution of the exponential equation, rounded to four decimal places. $$\frac{10}{1+e^{-x}}=2$$

Bird Population The population of a certain species of bird is limited by the type of habitat required for nesting. The population behaves according to the logistic growth model $$n(t)=\frac{5600}{0.5+27.5 e^{-0.044 t}}$$ where \(t\) is measured in years. (a) Find the initial bird population. (b) Draw a graph of the function \(n(t)\) (c) What size does the population approach as time goes on?

Use the Laws of Logarithms to expand the expression. $$\log _{6} \sqrt[4]{17}$$

Evaluate the expression. (a) \(\log _{8} 0.25\) (b) \(\ln e^{4}\) (c) \(\ln (1 / e)\)

These exercises use Newton’s Law of Cooling. A roasted turkey is taken from an oven when its temperature has reached \(185^{\circ} \mathrm{F}\) and is placed on a table in a room where the temperature is \(75^{\circ} \mathrm{F}\). (a) If the temperature of the turkey is \(150^{\circ} \mathrm{F}\) after half an hour, what is its temperature after 45 min? (b) When will the turkey cool to \(100^{\circ} \mathrm{F} ?\)

Find the solution of the exponential equation, rounded to four decimal places. $$100(1.04)^{2 t}=300$$

World Population The relative growth rate of world population has been decreasing steadily in recent years. On the basis of this, some population models predict that world population will eventually stabilize at a level that the planet can support. One such logistic model is $$P(t)=\frac{73.2}{6.1+5.9 e^{-0.021}}$$ where \(t=0\) is the year 2000 and population is measured in billions. (a) What world population does this model predict for the year \(2200 ?\) For \(2300 ?\) (b) Sketch a graph of the function \(P\) for the years 2000 to 2500. (c) According to this model, what size does the world population seem to approach as time goes on?

Use the Laws of Logarithms to expand the expression. $$\log _{3}(x \sqrt{y})$$

Evaluate the expression. (a) \(\log _{4} \sqrt{2}\) (b) \(\log _{4}\left(\frac{1}{2}\right)\) (c) \(\log _{4} 8\)

These exercises use Newton’s Law of Cooling. A kettle full of water is brought to a boil in a room with temperature \(20^{\circ} \mathrm{C}\). After 15 min the temperature of the water has decreased from \(100^{\circ} \mathrm{C}\) to \(75^{\circ} \mathrm{C}\). Find the temperature after another 10 min. Illustrate by graphing the temperature function.

Find the solution of the exponential equation, rounded to four decimal places. $$(1.00625)^{12 t}=2$$

Tree Diameter For a certain type of tree the diameter \(D\) (in feet) depends on the tree's age \(t\) (in years) according to the logistic growth model $$D(t)=\frac{5.4}{1+2.9 e^{-0.01 t}}$$ Find the diameter of a 20 -year-old tree. THE GRAPH CAN'T COPY .

Use the Laws of Logarithms to expand the expression. $$\log _{2}(x y)^{10}$$

Use the definition of the logarithmic function to find \(x .\) (a) \(\log _{2} x=5\) (b) \(\log _{2} 16=x\)

These exercises deal with logarithmic scales. The hydrogen ion concentration of a sample of each substance is given. Calculate the \(\mathrm{pH}\) of the substance. (a) Lemon juice: \(\left[\mathrm{H}^{+}\right]=5.0 \times 10^{-3} \mathrm{M}\) (b) Tomato juice: \(\left[\mathrm{H}^{+}\right]=3.2 \times 10^{-4} \mathrm{M}\) (c) Seawater: \(\left[\mathrm{H}^{+}\right]=5.0 \times 10^{-9} \mathrm{M}\)

Solve the equation. $$e^{2 x}-3 e^{x}+2=0$$

Compound Interest An investment of \(\$ 7,000\) is deposited into an account in which interest is compounded continuously. Complete the table by filling in the amounts to which the investment grows at the indicated times or interest rates. $$r=3 \%$$ $$\begin{array}{|c|c|} \hline \begin{array}{c} \text { Time } \\ \text { (years) } \end{array} & \text { Amount } \\ \hline 1 & \\ 2 & \\ 3 & \\ 4 & \\ 5 & \\ 6 & \\ \hline \end{array}$$

Use the Laws of Logarithms to expand the expression. $$\log _{5} \sqrt[3]{x^{2}+1}$$

Use the definition of the logarithmic function to find \(x .\) (a) \(\log _{5} x=4\) (b) \(\log _{10} 0.1=x\)

These exercises deal with logarithmic scales. An unknown substance has a hydrogen ion concentration of \(\left[\mathrm{H}^{+}\right]=3.1 \times 10^{-8} \mathrm{M}\). Find the \(\mathrm{pH}\) and classify the substance as acidic or basic.

Solve the equation. $$e^{2 x}-e^{x}-6=0$$

Compound Interest An investment of \(\$ 7,000\) is deposited into an account in which interest is compounded continuously. Complete the table by filling in the amounts to which the investment grows at the indicated times or interest rates. \(t=10\) years $$\begin{array}{|c|c|} \hline \begin{array}{c} \text { Rate } \\ \text { per year } \end{array} & \text { Amount } \\ \hline 1 \% & \\ 2 \% & \\ 3 \% & \\ 4 \% & \\ 5 \% & \\ 6 \% & \\ \hline \end{array}$$

Use the Laws of Logarithms to expand the expression. $$\log _{a}\left(\frac{x^{2}}{y z^{3}}\right)$$

Use the definition of the logarithmic function to find \(x .\) (a) \(\log _{3} 243=x\) (b) \(\log _{3} x=3\)

These exercises deal with logarithmic scales. The pH reading of a sample of each substance is given. Calculate the hydrogen ion concentration of the substance. (a) Vinegar: \(\mathrm{pH}=3.0\) (b) Milk: \(\mathrm{pH}=6.5\)

Solve the equation. $$e^{4 x}+4 e^{2 x}-21=0$$

Use the Laws of Logarithms to expand the expression. $$\ln \sqrt{a b}$$

Use the definition of the logarithmic function to find \(x .\) (a) \(\log _{4} 2=x\) (b) \(\log _{4} x=2\)

These exercises deal with logarithmic scales. The pH reading of a glass of liquid is given. Find the hydrogen ion concentration of the liquid. (a) Beer: \(\mathrm{pH}=4.6\) (b) Water: \(\mathrm{pH}=7.3\)

Solve the equation. $$e^{x}-12 e^{-x}-1=0$$

Use the Laws of Logarithms to expand the expression. $$\ln \sqrt[3]{3 r^{2} s}$$