Chapter 18

The approximate density of seawater at a depth of \(h\) mi is $$d=64.0 e^{0.00676 h}\left(\mathrm{lb} / \mathrm{ft}^{3}\right)$$ Find the depth \(h\) at which the density will be \(64.5 \mathrm{lb} / \mathrm{ft}^{3}.\)

Solve for \(x\). Give any approximate results to three significant digits. Check your answers. $$\ln (5 x+2)-\ln (x+6)=\ln 4$$

A certain radioactive material loses its radioactivity at the rate of \(2 \frac{1}{2} \%\) per year. What fraction of its initial radioactivity will remain after 10.0 years?

Solve for \(x\). Give any approximate results to three significant digits. Check your answers. $$\log x+\log 4 x=2$$

Simplify each expression. $$\log _{2} 2$$

Using the formula for compound interest, Eq. \(1009, y=a(1+n)^{t},\) calculate the number of years it will take a sum of money to triple when invested at a rate of \(12 \%\) per year.

Find the number whose common logarithm is given. $$0.366$$

Solve for \(x\). Give any approximate results to three significant digits. Check your answers. $$\ln x+\ln (x+2)=1$$

Simplify each expression. $$\log _{e} e$$

Find the number whose common logarithm is given. $$2.227$$

Solve for \(x\). Give any approximate results to three significant digits. Check your answers. $$\log 8 x^{2}-\log 4 x=2.54$$

Simplify each expression. $$\log _{10} 10$$

Find the half-life of a material that decays exponentially at the rate of \(3.50 \%\) per year.

In a certain fabric mill, cloth is removed from a dye bath and is then observed to dry exponentially at the rate of \(24 \%\) per hour. What percent of the original moisture will still be present after 5 h?

Solve for \(x\). Give any approximate results to three significant digits. Check your answers. $$2 \log x-\log (1-x)=1$$

Simplify each expression. $$\log _{3} 3^{2}$$

How long will it take the U.S. annual oil consumption to double if it is increasing exponentially at a rate of \(7.0 \%\) per year?

Simplify each expression. $$\log _{e} e^{x}$$

Computer: Assuming that the present annual world oil consumption is \(17 \times 10^{9}\) barrels/yr, that this rate of consumption is increasing at a rate of \(5 \%\) per year, and that the total world oil reserves are \(1700 \times 10^{9}\) barrels, compute and print the following table:$$\begin{array}{ccc}\hline \text { Year } & \text { Annual Consumption } & \text { Oil Remaining } \\\\\hline 0 & 17 & 1700 \\\1 & 17.85 & 1682.15 \\\& & \\\\\cdot & & \\ & &\end{array}$$,Have the computation stop when the oil reserves are almost gone.

How long will it take the world population to double at an exponential growth rate of \(1.64 \%\) per year?

Find the natural logarithm of each number. $$48.3$$

Simplify each expression. $$\log _{10} 10^{x}$$

Solve for \(x\). Give any approximate results to three significant digits. Check your answers. $$\log \left(x^{2}-4\right)-1=\log (x+2)$$

We can get an approximate value for \(e\) from the following infinite series:$$\begin{aligned} &\begin{array}{l}\text { Series } \\\\\text { Approximation } \quad e=2+\frac{1}{2 !}+\frac{1}{3 !}+\frac{1}{4 !}+\ldots \quad 158 \\\\\text { for }e\end{array}\\\&\end{aligned}$$.where \(4 !\) (read " 4 factorial") is \(4(3)(2)(1)=24\). Write a program to compute \(e\) using the first five terms of the series.

Find the natural logarithm of each number. $$846$$

Simplify each expression. $$e^{\log _{e} x}$$

Solve for \(x\). Give any approximate results to three significant digits. Check your answers. $$2 \log x-1=\log (20-2 x)$$

The infinite series often used to calculate \(e^{x}\) is as follows:$$\begin{array}{|lc|}\hline \text { Series } & \\\\\text { Approximation } & e^{x}=1+x+\frac{x^{2}}{2 !}+\frac{x^{3}}{3 !}+\frac{x^{4}}{4 !}+\ldots & 159 \\\\\text { for } e^{x} & & \\\\\hline\end{array}$$.Write a program to compute \(e^{x}\) by using the first 15 terms of this series, and use it to find \(e^{5}\).

Find the natural logarithm of each number. $$2365$$

Simplify each expression. $$2^{\log _{2} 3 y}$$

Solve for \(x\). Give any approximate results to three significant digits. Check your answers. $$\log \left(x^{2}-1\right)-2=\log (x+1)$$

Find the natural logarithm of each number. $$1.285$$

Simplify each expression. $$10^{\log x^{2}}$$

Solve for \(x\). Give any approximate results to three significant digits. Check your answers. $$\ln 2 x-\ln 4+\ln (x-2)=1$$

Change of Base Find the common logarithm of the number whose natural logarithm is the given value. $$8.36$$

Find the natural logarithm of each number. $$1.845$$

Change of Base Find the common logarithm of the number whose natural logarithm is the given value. $$-3.846$$

Find the natural logarithm of each number. $$4.77$$

Change of Base Find the common logarithm of the number whose natural logarithm is the given value. $$3.775$$

Find the natural logarithm of each number. $$1.374$$

Applications. Use Eq. \(1103, G_{p}=10 \log _{10} \frac{P_{2}}{P_{1}},\) to find the power transmitted by a transmission line with an input of \(2750 \mathrm{kW}\) and a loss of \(3.25 \mathrm{dB}\)

Change of Base Find the common logarithm of the number whose natural logarithm is the given value. $$15.36$$

Find the natural logarithm of each number. $$45,900$$

Find the natural logarithm of each number. $$1.364$$

Change of Base Find the common logarithm of the number whose natural logarithm is the given value. $$5.26$$

Find the number whose natural logarithm is given. $$2.846$$

Change of Base Find the common logarithm of the number whose natural logarithm is the given value. $$-0.638$$

Applications. \(p H:\) The \(\mathrm{pH}\) value of a solution having a concentration \(C\) of hydrogen ions is given by the following equation: $$\begin{array}{|llll} \text { pH } & \text { pH }=-\log _{10} C & 1048 \end{array}$$ The units of \(C\) are moles of hydrogen ions per liter of solution. A mole is a unit used in chemistry for expressing large quantities of very small entities. A mole of hydrogen ions, for example, has a mass of one gram. Find the concentration of hydrogen ions in a solution having a pH of 4.65.

Applications. A pH of 7 is considered neutral, while a lower \(\mathrm{pH}\) is acidic and a higher \(\mathrm{pH}\) is alkaline. What is the hydrogen ion concentration at a pH of \(7.0 ?\)

Find the number whose natural logarithm is given. $$0.879$$