Chapter 5

\(19-44\) Use the Laws of Logarithms to expand the expression. $$ \log _{5} \frac{x}{2} $$

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?

\(17-24\) . These exercises use the radioactive decay model. Carbon-14 Dating 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.)

Evaluate the expression. $$ \begin{array}{llll}{\text { (a) } \log _{2}\left(\frac{1}{n}\right)} & {\text { (b) } \log _{10} \sqrt{10}} & {\text { (c) } \log _{3} 0.2}\end{array} $$

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

\(19-44\) Use the Laws of Logarithms to expand the expression. $$ \log 6^{10} $$

Sky Diving A sky diver jumps from a reasonable height above the ground. The air resistance she experiences is proportional to her velocity, and the constant of proportionality is \(0.2 .\) It can be shown that the downward velocity of the sky diver at time \(t\) is given by $$ v(t)=80\left(1-e^{-0.2 t}\right) $$ where \(t\) is measured in seconds and \(v(t)\) is measured in feet per second \((f t / s)\) (a) Find the initial velocity of the sky diver. (b) Find the velocity after 5 \(\mathrm{s}\) and after 10 \(\mathrm{s}\) . (c) Draw a graph of the velocity function \(v(t)\) (d) The maximum velocity of a falling object with wind resistance is called its terminal velocity. From the graph in part (c) find the terminal velocity of this sky diver.

\(17-24\) . These exercises use the radioactive decay model. Carbon-14 Dating 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.)

Evaluate the expression. $$ \begin{array}{llll}{\text { (a) }\log _{3} 125} & {\text { (b) } \log _{49} 7} & {\text { (c) } \log _{8} \sqrt{3}}\end{array} $$

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

\(19-44\) Use the Laws of Logarithms to expand the expression. $$ \ln \sqrt{z} $$

Mixtures and Concentrations A 50 -gallon barrel is filled completely with pure water. Salt water with a concentration of 0.3 lb/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 \(Q(t) .\)

\(25-28=\) These exercises use Newton's Law of Cooling. Cooling Soup 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 \(^{\circ} \mathrm{F}\) . (a) What is the initial temperature of the soup? (b) What is the temperature after 10 \(\mathrm{min}\) ? (c) After how long will the temperature be \(100^{\circ} \mathrm{F} ?\)

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

Evaluate the expression. $$ \begin{array}{llll}{\text { (a) } 2^{\log _{2} 17}} & {\text { (b) } 3^{\ln 8}} & {\text { (c) } e^{\ln \sqrt{3}}}\end{array} $$

\(19-44\) Use the Laws of Logarithms to expand the expression. $$ \log _{2}\left(A B^{2}\right) $$

Logistic Growth Animal populations are not capable of unrestricted growth because of limited habitat and food supplies. Under such conditions the population follows a logistic growth model: $$ P(t)=\frac{d}{1+k e^{-c t}} $$ where \(c, d,\) and \(k\) are positive constants. For a certain fish population in a small pond \(d=1200, k=11, c=0.2,\) and \(t\) is measured in years. The fish were introduced into the pond at time \(t=0\) . (a) How many fish were originally put in the pond? (b) Find the population after \(10,20,\) and 30 years. (c) Evaluate \(P(t)\) for large values of \(t\) . What value does the population approach as \(t \rightarrow \infty\) Does the graph shown confirm your calculations?

\(25-28=\) These exercises use Newton's Law of Cooling. Time of Death 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 napproximately \(k=0.1947\) , assuming that time is measured in hours. Suppose that the temperature of the surroundings is \(60^{\circ} \mathrm{F}\) . (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. \(\frac{10}{1+e^{-x}}=2\)

Evaluate the expression. $$ \begin{array}{llll}{\text { (a) } e^{\ln \pi}} & {\text { (b) } 10^{\log 5}} & {\text { (c) } 10^{\log .87}}\end{array} $$

\(19-44\) Use the Laws of Logarithms to expand the expression. $$ \log _{6} \sqrt[4]{17} $$

\(25-28=\) These exercises use Newton's Law of Cooling. Cooling Turkey 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 \(\mathrm{min}\) ? (b) When will the turkey cool to \(100^{\circ} \mathrm{F} ?\)

\(19-44\) Use the Laws of Logarithms to expand the expression. $$ \log _{3}(x \sqrt{y}) $$

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 $$ P(t)=\frac{73.2}{6.1+5.9 e^{-0.02 t}} $$ where \(t=0\) is the year 2000 and population is measured in billions. 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?

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

Evaluate the expression. $$ \begin{array}{llll}{\text { (a) } \log _{4} \sqrt{2}} & {\text { (b) } \log _{4}\left(\frac{1}{2}\right)} & {\text { (c) } \log _{4} 8}\end{array} $$

\(19-44\) Use the Laws of Logarithms to expand the expression. $$ \log _{2}(X y)^{10} $$

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.

\(29-43\) . These exercises deal with logarithmic scales. Finding pH The hydrogen ion concentration of a sample of each substance is given. Calculate the 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. \(e^{2 x}-3 e^{x}+2=0\)

Use the definition of the logarithmic function to find \(x\) $$ \begin{array}{ll}{\text { (a) } \log _{2} x=5} & {\text { (b) } \log _{2} 16=x}\end{array} $$

\(19-44\) Use the Laws of Logarithms to expand the expression. $$ \log _{5} \sqrt[3]{x^{2}+1} $$

\(29-43\) . These exercises deal with logarithmic scales. Finding \(\mathrm{pH}\) An unknown substance has a hydrogen ion concentration of \(\left[\mathrm{H}^{+}\right]=3.1 \times 10^{-8} \mathrm{M} .\) Find the pH and classify the substance as acidic or basic.

Solve the equation. \(e^{2 x}-e^{x}-6=0\)

Use the definition of the logarithmic function to find \(x\) $$ \begin{array}{ll}{\text { (a) } \log _{3} x=4} & {\text { (b) } \log _{10} 0.1=x}\end{array} $$

\(19-44\) 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 \(x\). $$ \begin{array}{ll}{\text { (a) } \log _{3} 243=x} & {\text { (b) } \log _{3} x=3}\end{array} $$

\(29-43\) . These exercises deal with logarithmic scales. Ion Concentration The pH reading of a sample of each substance is given. Calculate the hydrogen ion concentration of the substance. (a) Vinegar: pH \(=3.0\) (b) Milk: \(\mathrm{pH}=6.5\)

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

\(19-44\) Use the Laws of Logarithms to expand the expression. $$ \ln \sqrt{a b} $$

Compound Interest If \(\$ 2000\) is invested at an interest rate of 3.5\(\%\) per year, compounded continuously, find the value of the investment after the given number of years. \(\begin{array}{llll}{\text { (a) } 2 \text { years }} & {\text { (b) } 4 \text { years }} & {\text { (c) } 12 \text { years }}\end{array}\)

Use the definition of the logarithmic function to find \(x\). $$ \begin{array}{ll}{\text { (a) } \log _{4} 2=x} & {\text { (b) } \log _{4} x=2}\end{array} $$

\(29-43\) . These exercises deal with logarithmic scales. Ion Concentration 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. \(e^{x}-12 e^{-x}-1=0\)

\(19-44\) Use the Laws of Logarithms to expand the expression. $$ \ln \sqrt[3]{3 r^{2} s} $$

Compound Interest If \(\$ 3500\) is invested at an interest rate of 6.25\(\%\) per year, compounded continuously, find the value of the investment after the given number of years. \(\begin{array}{llll}{\text { (a) } 3 \text { years }} & {\text { (b) } 6 \text { years }} & {\text { (c) } 9 \text { years }}\end{array}\)

Use the definition of the logarithmic function to find \(x\). $$ \begin{array}{ll}{\text { (a) } \log _{10} x=2} & {\text { (b) } \log _{5} x=2}\end{array} $$

\(29-43\) . These exercises deal with logarithmic scales. Finding pH The hydrogen ion concentrations in cheeses range from \(4.0 \times 10^{-7} \mathrm{M}\) to \(1.6 \times 10^{-5} \mathrm{M} .\) Find the corresponding range of pH readings.

Solve the equation. \(x^{2} 2^{x}-2^{x}=0\)

\(19-44\) Use the Laws of Logarithms to expand the expression. $$ \log \left(\frac{x^{3} y^{4}}{z^{6}}\right) $$