Chapter 5

Determine whether the function is even, odd, or neither . $$f(x)=\frac{e^{x}-e^{-x}}{2}$$

Translate the given exponential statement into an equivalent logarithmic statement. $$e^{k}=t$$

Write the given expression without using radicals. $$\frac{1}{\sqrt{x}}$$

Solve the equation for \(x\) by first making an appropriate substitution. $$9^{x}-4 \cdot 3^{x}+3=0$$

Use graphical or algebraic means to determine whether the statement is true or false. $$\ln (x+5)=\ln (x)+\ln 5 ?$$

Determine whether the function is even, odd, or neither . $$f(x)=e^{-x^{2}}$$

Translate the given exponential statement into an equivalent logarithmic statement. $$e^{2 / r}=w$$

The number of children who were home schooled in the United States in selected years is shown in the table. (a) Sketch a scatter plot of the data, with \(x=0\) corresponding to 1980 (b) Find a quadratic model for the data. (c) Find a logistic model for the data. (d) What is the number of home-schooled children predicted by each model for the year \(2015 ?\) (e) What are the limitations of each model? $$\begin{array}{|c|c|} \hline \begin{array}{c} \text { Fall of } \\ \text { School Year } \end{array} & \begin{array}{c} \text { Number of } \\ \text { Children (in thousands) } \end{array} \\ \hline 1985 & 183 \\ \hline 1988 & 225 \\ \hline 1990 & 301 \\ \hline 1992 & 470 \\ \hline 1993 & 588 \\ \hline 1994 & 735 \\ \hline 1995 & 800 \\ \hline 1996 & 920 \\ \hline 1997 & 1100 \\ \hline 1999 & 1400 \\ \hline 2000 & 1700 \\ \hline 2005 & 1900 \\ \hline \end{array}$$

Write the given expression without using radicals. $$\sqrt[5]{x^{2}}$$

Solve the equation for \(x\) by first making an appropriate substitution. $$25^{x}-8 \cdot 5^{x}=-12$$

Use the Big-Little Principle to explain why \(e^{x}+e^{-x}\) is approximately equal to \(e^{x}\) when \(x\) is large.

Translate the given exponential statement into an equivalent logarithmic statement. $$e^{e}=15.1543$$

Solve the equation for \(x\) by first making an appropriate substitution. $$e^{2 x}-5 e^{x}+6=0$$

Find the average rate of change of the function. \(f(x)=3\left(4^{x}\right)\) as \(x\) goes from 1 to 3

Evaluate the given expression without using a calculator. $$\log 10^{\sqrt{43}}$$

Write the given expression without using radicals. $$\sqrt{\sqrt[3]{a^{3} b^{4}}}$$

Solve the equation for \(x\) by first making an appropriate substitution. $$3 e^{2 x}-16 e^{x}+5=0$$

If \(\ln b^{10}=10,\) what is \(b ?\)

Evaluate the given expression without using a calculator. $$\log 10^{\sqrt[3]{r^{2}-s^{2}}}$$

Find the average rate of change of the function. \(f(x)=3\left(4^{x}\right)\) as \(x\) goes from 10 to 12

Write the given expression without using radicals. $$\sqrt[5]{t} \sqrt{16 t^{5}}$$

Solve the equation for \(x\) by first making an appropriate substitution. $$6 e^{2 x}-16 e^{x}=6$$

Prove the Quotient Law for Logarithms: For \(v, w>0\) \(\ln \left(\frac{v}{w}\right)=\ln v-\ln w .\) (Use properties of exponents and the fact that \(\left.v=e^{\ln v} \text { and } w=e^{\ln w} .\right)\)

Evaluate the given expression without using a calculator. $$\ln e^{15}$$

Find the average rate of change of the function. \(g(x)=3^{x^{2}-x-3}\) as \(x\) goes from -1 to 1

In the past two decades, more women than men have been entering college. The table shows the percentage of male first-year college students in selected years. $$\begin{array}{|l|l|l|l|l|l|l|l|l|l|} \hline \text { Year } & 1985 & 1990 & 1995 & 1997 & 1998 & 1999 & 2003 & 2004 & 2005 \\ \hline \text { Percent } & 48.9 & 46.9 & 45.6 & 45.5 & 45.5 & 45.3 & 45.1 & 44.9 & 45.0 \\ \hline \end{array}$$ (a) Find three models for this data: exponential, logarithmic, and power, with \(x=5\) corresponding to 1985 (b) For the years \(1985-2005,\) is there any significant difference among the models? (c) Assume that the models remain accurate. What year does each predict as the first year in which fewer than \(43 \%\) of first-year college students will be male? (d) We actually have some additional data: $$\begin{array}{|l|c|c|c|c|} \hline \text { Year } & 2000 & 2001 & 2002 & 2006 \\ \hline \text { Percent } & 45.2 & 44.9 & 45.0 & 45.1 \\ \hline \end{array}$$ Which model did the best job of predicting the new data?

Write the given expression without using radicals. $$\sqrt{x}(\sqrt[3]{x^{2}})(\sqrt[4]{x^{3}})$$

Solve the equation for \(x\) by first making an appropriate substitution. $$6 e^{2 x}+7 e^{x}=10$$

'Find the average rate of change of the function. \(h(x)=2^{x}\) as \(x\) goes from 1 to 2

Evaluate the given expression without using a calculator. $$e^{\ln \pi}$$

Simplify the expression without using a calculator. $$\sqrt{80}$$

Solve the equation for \(x\) by first making an appropriate substitution. $$4^{x}+6 \cdot 4^{-x}=5$$

Find the average rate of change of the function. \(h(x)=2^{x}\) as \(x\) goes from 1 to 1.001

Evaluate the given expression without using a calculator. $$\ln \sqrt{e}$$

The table gives the death rate in motor vehicle accidents (per 100,000 population) in selected years. $$\begin{array}{|l|c|c|c|c|c|c|c|} \hline \text { Year } & 1970 & 1980 & 1985 & 1990 & 1995 & 2000 & 2003 \\ \hline \text { Death Rate } & 26.8 & 23.4 & 19.3 & 18.8 & 16.5 & 15.6 & 15.4 \\\ \hline \end{array}$$ (a) Find an exponential model for the data, with \(x=0\) corresponding to 1970 . (b) What was the death rate in 1998 and in \(2002 ?\) (c) Assume that the model remains accurate, when will the death rate drop to 13 per \(100,000 ?\)

Simplify the expression without using a calculator. $$\sqrt{120}$$

State the magnitude on the Richter scale of an earthquake that satisfies the given condition. 250 times stronger than the zero quake.

Evaluate the given expression without using a calculator. $$\ln \sqrt[5]{e}$$

Simplify the expression without using a calculator. $$\sqrt{6} \sqrt{12}$$

Solve the equation for \(x\). $$\frac{e^{x}-e^{-x}}{2}=t$$

Find the average rate of change of the function. \(h(x)=e^{x}\) as \(x\) goes from 1 to 1.001

Evaluate the given expression without using a calculator. $$e^{\ln 931}$$

Simplify the expression without using a calculator. $$\sqrt[3]{12} \sqrt[3]{10}$$

Solve the equation for \(x\). $$\frac{e^{x}-e^{-x}}{e^{x}+e^{-x}}=t$$

Deal with the energy intensity i of a sound, which is related to the loudness of the sound by the function \(L(i)=10 \cdot \log \left(i / i_{0}\right),\) where \(i_{0}\) is the minimum intensity detectable by the human ear and \(L(i)\) is measured in decibels. Find the decibel measure of the sound. Ticking watch (intensity is 100 times \(i_{0}\) ).

Find the average rate of change of the function. \(f(x)=a^{x}, a>0,\) as \(x\) goes from 0 to 0.001

Evaluate the given expression without using a calculator. $$\log (\log (10,000,000,000))$$

Simplify the expression without using a calculator. $$\frac{-6+\sqrt{99}}{15}$$

(a) Prove that if \(\ln u=\ln v,\) then \(u=v .\) (b) Is it always the case that if \(u=v\) then \(\ln u=\ln v ?\) Why or why not?

Deal with the energy intensity i of a sound, which is related to the loudness of the sound by the function \(L(i)=10 \cdot \log \left(i / i_{0}\right),\) where \(i_{0}\) is the minimum intensity detectable by the human ear and \(L(i)\) is measured in decibels. Find the decibel measure of the sound. Soft music (intensity is 10,000 times \(i_{0}\) ).