Chapter 5

Compute the ratios of successive entries in the table to determine whether or not an exponential model is appropriate for the data. $$\begin{array}{|l|l|l|l|l|l|l|} \hline x & 1 & 3 & 5 & 7 & 9 & 11 \\ \hline y & 3 & 21 & 55 & 105 & 171 & 253 \\ \hline \end{array}$$

Compute and simplify. $$x^{1 / 2}\left(3 x^{3 / 2}+2 x^{-1 / 2}\right)$$

Solve the equation. First express your answer in terms of natural logarithms (for instance, \(x=(2+\ln 5) /(\ln 3)) .\) Then use a calculator to find an approximation for the answer. $$9^{x-1}=8^{x-3}$$

In Exercises \(11-16,\) let \(u=\ln x\) and \(v=\ln y .\) Write the given expression in terms of u and v. For example, $$\ln x^{3} y=\ln x^{3}+\ln y=3 \ln x+\ln y=3 u+v$$ $$\ln \left(x^{4} y^{3}\right)$$

Translate the given logarithmic statement into an equivalent exponential statement. $$\ln s=r$$

List the transformations needed to transform the graph of \(h(x)=2^{x}\) into the graph of the given function. $$g(x)=-\left(2^{x}\right)$$

Compute and simplify. $$\left(x^{1 / 2}+y^{1 / 2}\right)\left(x^{1 / 2}-y^{1 / 2}\right)$$

In Exercises \(11-16,\) let \(u=\ln x\) and \(v=\ln y .\) Write the given expression in terms of u and v. For example, $$\ln x^{3} y=\ln x^{3}+\ln y=3 \ln x+\ln y=3 u+v$$ $$\ln \left(\sqrt{x} \cdot y^{2}\right)$$

List the transformations needed to transform the graph of \(h(x)=2^{x}\) into the graph of the given function. $$k(x)=3\left(2^{x}\right)$$

Compute and simplify. $$\left(x^{1 / 3}+y^{1 / 2}\right)\left(2 x^{1 / 3}-y^{3 / 2}\right)$$

In Exercises \(11-16,\) let \(u=\ln x\) and \(v=\ln y .\) Write the given expression in terms of u and v. For example, $$\ln x^{3} y=\ln x^{3}+\ln y=3 \ln x+\ln y=3 u+v$$ $$\ln \left(\frac{\sqrt{x y}}{y^{2}}\right)$$

Translate the given logarithmic statement into an equivalent exponential statement. $$\log (a+c)=d$$

List the transformations needed to transform the graph of \(h(x)=2^{x}\) into the graph of the given function. $$g(x)=2^{x-1}$$

Graph each of the following power functions in a window with \(0 \leq x \leq 20\) (a) \(f(x)=x^{-1.5}\) (b) \(g(x)=x^{75}\) (c) \(h(x)=x^{2.4}\)

Compute and simplify. $$(x+y)^{1 / 2}\left[(x+y)^{1 / 2}-(x+y)\right]$$

Solve the equation. First express your answer in terms of natural logarithms (for instance, \(x=(2+\ln 5) /(\ln 3)) .\) Then use a calculator to find an approximation for the answer. $$2^{1-3 x}=3^{x+1}$$

In Exercises \(11-16,\) let \(u=\ln x\) and \(v=\ln y .\) Write the given expression in terms of u and v. For example, $$\ln x^{3} y=\ln x^{3}+\ln y=3 \ln x+\ln y=3 u+v$$ $$\ln (\sqrt[3]{x^{2} \sqrt{y}})$$

Translate the given exponential statement into an equivalent logarithmic statement. $$10^{-2}=.01$$

List the transformations needed to transform the graph of \(h(x)=2^{x}\) into the graph of the given function. $$f(x)=2^{x+2}-5$$

Compute and simplify. $$\left(x^{1 / 3}+y^{1 / 3}\right)\left(x^{2 / 3}-x^{1 / 3} y^{1 / 3}+y^{2 / 3}\right)$$

In Exercises \(11-16,\) let \(u=\ln x\) and \(v=\ln y .\) Write the given expression in terms of u and v. For example, $$\ln x^{3} y=\ln x^{3}+\ln y=3 \ln x+\ln y=3 u+v$$ $$\ln \left(\frac{\sqrt[3]{x^{2} y^{2}}}{x^{5}}\right)$$

Translate the given exponential statement into an equivalent logarithmic statement. $$10^{3}=1000$$

List the transformations needed to transform the graph of \(h(x)=2^{x}\) into the graph of the given function. $$g(x)=-5\left(2^{x-1}\right)+7$$

Determine whether an exponential, power, or logarithmic model (or none or several of these) is appropriate for the data by determining which (if any) of the following sets of points are approximately linear: $$\\{(x, \ln y)\\}, \quad\\{(\ln x, \ln y)\\}, \quad\\{(\ln x, y)\\}$$ where the given data set consists of the points \(\\{(x, y)\\}\) $$\begin{array}{|l|c|c|c|c|c|c|} \hline x & 1 & 3 & 5 & 7 & 9 & 11 \\ \hline y & 2 & 25 & 81 & 175 & 310 & 497 \\ \hline \end{array}$$

Factor the given expression. For example, $$x-x^{1 / 2}-2=\left(x^{1 / 2}-2\right)\left(x^{1 / 2}+1\right)$$ $$x^{2 / 3}+x^{1 / 3}-6$$

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

Solve the equation. First express your answer in terms of natural logarithms (for instance, \(x=(2+\ln 5) /(\ln 3)) .\) Then use a calculator to find an approximation for the answer. $$e^{2 x}=5$$

Determine whether an exponential, power, or logarithmic model (or none or several of these) is appropriate for the data by determining which (if any) of the following sets of points are approximately linear: $$\\{(x, \ln y)\\}, \quad\\{(\ln x, \ln y)\\}, \quad\\{(\ln x, y)\\}$$ where the given data set consists of the points \(\\{(x, y)\\}\) $$\begin{array}{|l|c|c|c|c|c|c|} \hline x & 3 & 6 & 9 & 12 & 15 & 18 \\ \hline y & 385 & 74 & 14 & 2.75 & .5 & .1 \\ \hline \end{array}$$

Factor the given expression. For example, $$x-x^{1 / 2}-2=\left(x^{1 / 2}-2\right)\left(x^{1 / 2}+1\right)$$ $$x^{2 / 7}-2 x^{1 / 7}-15$$

Solve the equation. First express your answer in terms of natural logarithms (for instance, \(x=(2+\ln 5) /(\ln 3)) .\) Then use a calculator to find an approximation for the answer. $$e^{-9 x}=3$$

Translate the given exponential statement into an equivalent logarithmic statement. $$10^{3 k}=6 r$$

Determine whether an exponential, power, or logarithmic model (or none or several of these) is appropriate for the data by determining which (if any) of the following sets of points are approximately linear: $$\\{(x, \ln y)\\}, \quad\\{(\ln x, \ln y)\\}, \quad\\{(\ln x, y)\\}$$ where the given data set consists of the points \(\\{(x, y)\\}\) $$\begin{array}{|l|c|c|c|c|c|c|} \hline x & 5 & 10 & 15 & 20 & 25 & 30 \\ \hline y & 17 & 27 & 35 & 40 & 43 & 48 \\ \hline \end{array}$$

Factor the given expression. For example, $$x-x^{1 / 2}-2=\left(x^{1 / 2}-2\right)\left(x^{1 / 2}+1\right)$$ $$x+4 x^{1 / 2}+3$$

Use graphical or algebraic means to determine whether the statement is true or false. $$\log x^{5}=5(\log x) ?$$

Solve the equation. First express your answer in terms of natural logarithms (for instance, \(x=(2+\ln 5) /(\ln 3)) .\) Then use a calculator to find an approximation for the answer. $$6 e^{-1.4 x}=21$$

Determine whether the function is even, odd, or neither . $$f(x)=10^{x}$$

Translate the given exponential statement into an equivalent logarithmic statement. $$e^{3.25}=25.79$$

Determine whether an exponential, power, or logarithmic model (or none or several of these) is appropriate for the data by determining which (if any) of the following sets of points are approximately linear: $$\\{(x, \ln y)\\}, \quad\\{(\ln x, \ln y)\\}, \quad\\{(\ln x, y)\\}$$ where the given data set consists of the points \(\\{(x, y)\\}\) $$\begin{array}{|l|c|c|c|c|c|c|} \hline x & 5 & 10 & 15 & 20 & 25 & 30 \\ \hline y & 2 & 110 & 460 & 1200 & 2500 & 4525 \\ \hline \end{array}$$

Factor the given expression. For example, $$x-x^{1 / 2}-2=\left(x^{1 / 2}-2\right)\left(x^{1 / 2}+1\right)$$ $$x^{1 / 3}+11 x^{1 / 6}+24$$

Use graphical or algebraic means to determine whether the statement is true or false. $$e^{x \ln x}=x^{x} \quad(x>0) ?$$

Solve the equation. First express your answer in terms of natural logarithms (for instance, \(x=(2+\ln 5) /(\ln 3)) .\) Then use a calculator to find an approximation for the answer. $$27 e^{-x / 4}=67.5$$

Translate the given exponential statement into an equivalent logarithmic statement. $$e^{3.14}=23.1039$$

The table shows the number of babies born as twins, triplets, quadruplets, etc., over a 7 -year period. $$\begin{array}{|l|c|} \hline \text { Year } & \text { Multiple Births } \\ \hline 1989 & 92,916 \\ \hline 1990 & 96,893 \\ \hline 1991 & 98,125 \\ \hline 1992 & 99,255 \\ \hline 1993 & 100,613 \\ \hline 1994 & 101,658 \\ \hline 1995 & 101,709 \\ \hline \end{array}$$ (a) Sketch a scatter plot of the data, with \(x=1\) corresponding to 1989 (b) Plot each of the following models on the same screen as the scatter plot. $$\begin{array}{l} f(x)=93,201.973+4,545.977 \ln x \\ g(x)=\frac{102,519.98}{1+.1536 e^{-4263 x}} \end{array}$$ (c) Use the table feature to estimate the number of multiple births in 2000 and 2010 . (d) Over the long run, which model do you think is the better predictor?

Factor the given expression. For example, $$x-x^{1 / 2}-2=\left(x^{1 / 2}-2\right)\left(x^{1 / 2}+1\right)$$ $$x^{4 / 5}-81$$

Use graphical or algebraic means to determine whether the statement is true or false. $$\ln x^{3}=(\ln x)^{3} ?$$

Solve the equation. First express your answer in terms of natural logarithms (for instance, \(x=(2+\ln 5) /(\ln 3)) .\) Then use a calculator to find an approximation for the answer. $$2.1 e^{(x / 2) \ln 3}=5$$

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. $$e^{12 / 7}=5.5527$$

Factor the given expression. For example, $$x-x^{1 / 2}-2=\left(x^{1 / 2}-2\right)\left(x^{1 / 2}+1\right)$$ $$x+3 x^{2 / 3}+3 x^{1 / 3}+1$$

Solve the equation. First express your answer in terms of natural logarithms (for instance, \(x=(2+\ln 5) /(\ln 3)) .\) Then use a calculator to find an approximation for the answer. $$2.7 e^{(-x / 3) \ln 7}=21$$