Chapter 5

Simplify the expression without a calculator $$ 2^{-3} $$

State the inverse action or actions. Opening a window

Concepts $$ \text { If } f(3)=2 \text { and } g(3)=5,(f+g)(3)=_______. $$

Simplify the expression without a calculator $$ (-3)^{-2} $$

Exercises \(I-6:\) (Refer to Example I.) Use a calculator to approximate each pair of expressions. Then state which property of logarithms this calculation illustrates. $$ \ln 12+\ln 5, \quad \ln 60 $$

State the inverse action or actions. Climbing up a ladder

Concepts $$ \text { If } f(3)=2 \text { and } g(2)=5,(g \circ f)(3)=______. $$

Evaluate each expression by hand, if possible. (a) \(\log (-3)\) (b) \(\log \frac{1}{100}\) (c) \(\log \sqrt{0.1}\) (d) \(\log 5^{\circ}\)

Simplify the expression without a calculator $$ 3(4)^{1 / 2} $$

State the inverse action or actions. Walking into a classroom, sitting down, and opening a book

Concepts $$ \text { If } f(x)=x^{2} \text { and } g(x)=4 x,(f g)(x)=______. $$

Evaluate each expression by hand, if possible. (a) \(\log 10,000\) (b) \(\log (-\pi)\) (c) \(\log \sqrt{0,001}\) (d) \(\log 8^{\circ}\)

Simplify the expression without a calculator $$ 5\left(\frac{1}{2}\right)^{-3} $$

State the inverse action or actions. Opening the door and turning on the lights

Concepts $$ \text { If } f(x)=x^{2} \text { and } g(x)=4 x,(f \circ g)(x)=_______. $$

Make a scatterplot of the data. Then find an exponential, logarithmic, or logistic function \(f\) that best models the data. $$\begin{array}{ccccc}x & 1 & 2 & 3 & 4 \\\\\hline y & 2.04 & 3.47 & 5.90 & 10.02\end{array}$$

Evaluate each expression by hand, if possible. (a) \(\log 10\) (b) \(\log 10,000\) (c) \(20 \log 0.1\) (d) \(\log 10+\log 0.001\)

Simplify the expression without a calculator $$ -2(27)^{2 / 3} $$

If \(f(x)\) calculates the number of square feet in \(x\) square yards and \(g(x)\) calculates the cost in dollars of \(x\) square feet of carpet, what does \((g \circ f)\) calculate?

Make a scatterplot of the data. Then find an exponential, logarithmic, or logistic function \(f\) that best models the data. $$\begin{array}{cccccc} x & 1 & 2 & 3 & 4 & 5 \\\\\hline y & 1.98 & 2.35 & 2.55 & 2.69 & 2.80 \end{array}$$

Evaluate each expression by hand, if possible. (a) \(\log 100\) (b) \(\log 1,000,000\) (c) \(5 \log 0.01\) (d) \(\log 0.1-\log 1000\)

Simplify the expression without a calculator $$ -4(8)^{-2 / 3} $$

Time Conversion If \(f(x)\) calculates the number of days in \(x\) hours and \(g(x)\) calculates the number of years in \(x\) days, what does \((g \circ f)\) calculate?

Make a scatterplot of the data. Then find an exponential, logarithmic, or logistic function \(f\) that best models the data. $$\begin{array}{cccccc} x & 1 & 2 & 3 & 4 & 5 \\ \hline y & 1.1 & 3.1 & 4.3 & 5.2 & 5.8 \end{array}$$

Evaluate each expression by hand, if possible. (a) \(2 \log 0.1+4\) (b) \(\log 10^{1 / 2}\) (c) \(3 \log 100-\log 1000\) (d) \(\log (-10)\)

Simplify the expression without a calculator $$ 4^{1 / 6} 4^{1 / 3} $$

Describe verbally the inverse of the statement. Then express both the given statement and its inverse symbolically. Subtract 2 from \(x\) and multiply the result by 3 .

Exercises \(7-10:\) Use \(f(x)\) and \(g(x)\) to evaluate each expression symbolically. \(f(x)=2 x-3, g(x)=1-x^{2}\) (a) \((f+g)(3)\) (b) \((f-g)(-1)\) (c) \((f g)(0)\) (d) \((f / g)(2)\)

Make a scatterplot of the data. Then find an exponential, logarithmic, or logistic function \(f\) that best models the data. $$\begin{array}{ccccccc} x & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline y & 1 & 2 & 4 & 7 & 9 & 10 \end{array}$$

Evaluate each expression by hand, if possible. (a) \(\log (-4)\) (b) \(\log 1\) (c) \(\log 0\) (d) \(-6 \log 100\)

Simplify the expression without a calculator $$ \frac{95 \sqrt{6}}{9^{1 / 3}} $$

Exercises \(7-10:\) Use \(f(x)\) and \(g(x)\) to evaluate each expression symbolically. \(f(x)=4 x-x^{3}, g(x)=x+3\) (a) \((g+g)(-2)\) (b) \((f-g)(0)\) (c) \((g f)(1)\) (d) \((g / f)(-3)\)

Make a scatterplot of the data. Then find an exponential, logarithmic, or logistic function \(f\) that best models the data. $$\begin{array}{ccccccc} x & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline y & 0.3 & 1.3 & 4.0 & 7.5 & 9.3 & 9.8\end{array}$$

Determine mentally an integer \(n\) so that the logarithm is between \(n\) and \(n+1 .\) Check your result with a calculator. (a) \(\log 79\) (b) \(\log 500\) (c) \(\log 5\) (d) \(\log 0.5\)

Simplify the expression without a calculator $$ e^{x} e^{x} $$

Exercises \(7-10:\) Use \(f(x)\) and \(g(x)\) to evaluate each expression symbolically. \(f(x)=2 x+1, g(x)=\frac{1}{x}\) (a) \((f+g)(2)\) (b) \((f-g)\left(\frac{1}{2}\right)\) (c) \((f g)(4)\) (d) \((f / g)(0)\)

Make a scatterplot of the data. Then find an exponential, logarithmic, or logistic function \(f\) that best models the data. $$\begin{array}{cccccc} x & 1 & 2 & 3 & 4 & 5 \\ y & 2.0 & 1.6 & 1.3 & 1.0 & 0.82 \end{array}$$

Determine mentally an integer \(n\) so that the logarithm is between \(n\) and \(n+1 .\) Check your result with a calculator. (a) \(\log 63\) (b) \(\log 5000\) (c) \(\log 9\) (d) \(\log 0.04\)

Simplify the expression without a calculator $$ e^{3 x} e^{1+x} $$

Describe verbally the inverse of the statement. Then express both the given statement and its inverse symbolically. Multiply \(x\) by \(-2\) and add 3.

Exercises \(7-10:\) Use \(f(x)\) and \(g(x)\) to evaluate each expression symbolically. \(f(x)=\sqrt[3]{x^{2}}, g(x)=|x-3|\) (a) \((f+g)(-8)\) (b) \((f-g)(-1)\) (c) \((f g)(0)\) (d) \((f / g)(27)\)

The table contains heart disease death rates per \(100,000\) people for selected ages. $$\begin{array}{|rccccc|}\hline \text { Age } & 30 & 40 & 50 & 60 & 70 \\\ \hline \text { Death rate } & 30.5 & 108.2 & 315 & 776 & 2010 \\ \hline \end{array}$$ (a) Make a scatterplot of the data in the viewing rectangle \([25,75,5]\) by \([-100,2100,200]\) (b) Find a function \(f\) that models the data. (c) Estimate the heart disease death rate for people who are 80 years old.

Find the exact value of each expression. (a) \(\log \sqrt{1000}\) (b) \(\log \sqrt[3]{10}\) (c) \(\log \sqrt[5]{0.1}\) (d) \(\log \sqrt{0.01}\)

Simplify the expression without a calculator $$ 3^{0} $$

Describe verbally the inverse of the statement. Then express both the given statement and its inverse symbolically. Take the reciprocal of a nonzero number \(x\).

The number of females working in automotive repair is increasing. The table shows the number of female ASE-certified technicians for selected years. $$\begin{array}{ccccc}\text { Year } & 1988 & 1989 & 1990 & 1991 \\ \hline \text { Total } & 556 & 614 & 654 & 737\end{array}$$ $$\begin{array}{|ccccc}\hline \text { Year } & 1992 & 1993 & 1994 & 1995 \\\\\hline \text { Total } & 849 & 1086 & 1329 & 1592\end{array}$$ (a) What type of function might model these data? (b) Use least-squares regression to find an exponential function given by \(f(x)=a b^{x}\) that models the data. Let \(x=0\) correspond to 1988 (c) Use \(f\) to estimate the number of certified female technicians in \(2005 .\) Round the result to the nearest hundred.

Find the exact value of each expression. (a) \(\log \sqrt{100,000}\) (b) \(\log \sqrt[3]{100}\) (c) \(2 \log \sqrt{0.1}\) (d) \(10 \log \sqrt[3]{10}\)

Simplify the expression without a calculator $$ 5\left(\frac{3}{4}\right)^{0} $$

Describe verbally the inverse of the statement. Then express both the given statement and its inverse symbolically. Take the square root of a positive number \(x\).

Exercises \(11-30:\) Use \(f(x)\) and \(g(x)\) to find a formula for each expression. Identify its domain. (a) \((f+g)(x)\) (b) \((f-g)(x)\) (c) \((f g)(x)\) (d) \((f / g)(x)\) $$ f(x)=1-4 x, \quad g(x)=3 x+1 $$