Chapter 5

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 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, correct to four decimal places. $$ \frac{10}{1+e^{-x}}=2 $$

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

\(15-24\) Evaluate the expression. $$ \begin{array}{llll}{\text { (a) } \log _{4} \sqrt{2}} & {\text { (b) } \log _{4}\left(\frac{1}{2}\right)} & {} & {\text { (c) } \log _{4} 8}\end{array} $$

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

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

\(25-32\) Use the definition of the logarithmic function to find \(x\). $$ \begin{array}{ll}{\text { (a) } \log _{2} x=5} & {\text { (b) } \log _{2} 16=x}\end{array} $$

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

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

\(25-32\) Use the definition of the logarithmic function to find \(x\). $$ \begin{array}{ll}{\text { (a) } \log _{5} x=4} & {\text { (b) } \log _{10} 0.1=x}\end{array} $$

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. $$ x^{2} 2^{x}-2^{x}=0 $$

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

\(25-32\) Use the definition of the logarithmic function to find \(x\). $$ \begin{array}{ll}{\text { (a) } \log _{3} 243=x} & {\text { (b) } \log _{3} x=3}\end{array} $$

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. $$ x^{2} 10^{x}-x 10^{x}=2\left(10^{x}\right) $$

Use the Laws of Logarithms to expand the expression. $$ \log \left(\frac{a^{2}}{b^{4} \sqrt{c}}\right) $$

\(25-32\) Use the definition of the logarithmic function to find \(x\). $$ \begin{array}{ll}{\text { (a) } \log _{4} 2=x} & {\text { (b) } \log _{4} x=2}\end{array} $$

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. $$ 4 x^{3} e^{-3 x}-3 x^{4} e^{-3 x}=0 $$

Use the Laws of Logarithms to expand the expression. $$ \log _{2}\left(\frac{x\left(x^{2}+1\right)}{\sqrt{x^{2}-1}}\right) $$

\(25-32\) Use the definition of the logarithmic function to find \(x\). $$ \begin{array}{ll}{\text { (a) } \log _{10} x=2} & {\text { (b) } \log _{5} x=2}\end{array} $$

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. $$ x^{2} e^{x}+x e^{x}-e^{x}=0 $$

Use the Laws of Logarithms to expand the expression. $$ \log _{5} \sqrt{\frac{x-1}{x+1}} $$

\(25-32\) Use the definition of the logarithmic function to find \(x\). $$ \begin{array}{lll}{\text { (a) } \log _{x} 1000=3} & {\text { (b) } \log _{x} 25=2}\end{array} $$

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 \(\mathrm{pH}\) readings.

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

Use the Laws of Logarithms to expand the expression. $$ \ln \left(x \sqrt{\frac{y}{z}}\right) $$

\(25-32\) Use the definition of the logarithmic function to find \(x\). $$ \begin{array}{ll}{\text { (a) } \log _{x} 16=4} & {\text { (b) } \log _{x} 8=\frac{3}{2}}\end{array} $$

The pH readings for wines vary from 2.8 to 3.8. Find the corresponding range of hydrogen ion concentrations.

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

Use the Laws of Logarithms to expand the expression. $$ \ln \frac{3 x^{2}}{(x+1)^{10}} $$

\(25-32\) Use the definition of the logarithmic function to find \(x\). $$ \begin{array}{ll}{\text { (a) } \log _{x} 6=\frac{1}{2}} & {\text { (b) } \log _{x} 3=\frac{1}{3}}\end{array} $$

If one earthquake is 20 times as intense as another, how much larger is its magnitude on the Richter scale?

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

Use the Laws of Logarithms to expand the expression. $$ \log \sqrt[4]{x^{2}+y^{2}} $$

\(33-36\) Use a calculator to evaluate the expression, correct to four decimal places. $$ \begin{array}{llll}{\text { (a) } \log 2} & {\text { (b) } \log 35.2} & {} & {\text { (c) } \log \left(\frac{2}{3}\right)}\end{array} $$

The 1906 earthquake in San Francisco had a magnitude of 8.3 on the Richter scale. At the same time in Japan an earthquake with magnitude 4.9 caused only minor damage. How many times more intense was the San Francisco earthquake than the Japanese earthquake?

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

Use the Laws of Logarithms to expand the expression. $$ \log \left(\frac{x}{\sqrt[3]{1-x}}\right) $$

\(33-36\) Use a calculator to evaluate the expression, correct to four decimal places. $$ \begin{array}{llll}{\text { (a) } \log 50} & {\text { (b) } \log \sqrt{2}} & {} & {\text { (c) } \log (3 \sqrt{2})}\end{array} $$

Solve the logarithmic equation for \(x\) $$ \ln x=10 $$

Use the Laws of Logarithms to expand the expression. $$ \log \sqrt{\frac{x^{2}+4}{\left(x^{2}+1\right)\left(x^{3}-7\right)^{2}}} $$

\(33-36\) Use a calculator to evaluate the expression, correct to four decimal places. $$ \begin{array}{llll}{\text { (a) } \ln 5} & {\text { (b) } \ln 25.3} & {\text { (c) } \ln (1+\sqrt{3})}\end{array} $$

The Northridge, California, earthquake of 1994 had a magnitude of 6.8 on the Richter scale. A year later, a 7.2-magnitude earthquake struck Kobe, Japan. How many times more intense was the Kobe earthquake than the Northridge earthquake?

Solve the logarithmic equation for \(x\) $$ \ln (2+x)=1 $$

Use the Laws of Logarithms to expand the expression. $$ \log \sqrt{x \sqrt{y \sqrt{z}}} $$

\(33-36\) Use a calculator to evaluate the expression, correct to four decimal places. $$ \begin{array}{llll}{\text { (a) } \ln 27} & {\text { (b) } \ln 7.39} & {\text { (c) } \ln 54.6} & {\text { (c) } \ln 54.6}\end{array} $$

The 1985 Mexico City earthquake had a magnitude of 8.1 on the Richter scale. The 1976 earthquake in Tangshan, China, was 1.26 times as intense. What was the magnitude of the Tangshan earthquake?