Chapter 4

Find the domain of each function. Use your answer to help you graph the function, and label all asymptotes. $$f(x)=4 \log _{3} x$$

Solve the logarithmic equation and eliminate any extraneous solutions. If there are no solutions, so state. $$\ln x=(x-2)^{2}$$

In Exercises \(53-64,\) simplify each expression. Assume that each variable expression is defined for appropriate values of \(x .\) Do not use a calculator. $$e^{\ln (2 x+1)}$$

Consider the function \(f(x)=e^{-x^{2}}.\) (a) Use a graphing utility to graph this function, with \(x\) ranging from -5 to 5 . You may need to scroll through the table of values to set an appropriate scale for the vertical axis. (b) What are the domain and range of \(f ?\) (c) Does \(f\) have any symmetries? (d) What are the \(x\) - and \(y\) -intercepts, if any, of the graph of this function? (e) Describe the behavior of the function as \(x\) approaches \(\pm \infty.\)

Find the domain of each function. Use your answer to help you graph the function, and label all asymptotes. $$f(x)=3 \log _{5} x$$

Determine how long it takes for the given investment to double if \(r\) is the interest rate and the interest is compounded continuously. Assume that no withdrawals or further deposits are made. Initial amount: \(\$ 1500 ; r=6 \%\)

In Exercises \(53-64,\) simplify each expression. Assume that each variable expression is defined for appropriate values of \(x .\) Do not use a calculator. $$\log _{2} 8$$

For an initial deposit of \(\$ 1500\) , find the total amount in a bank account after 5 years for the interest rates and compounding frequencies given. 6% compounded annually

Find the domain of each function. Use your answer to help you graph the function, and label all asymptotes. $$g(x)=\log x-3$$

Determine how long it takes for the given investment to double if \(r\) is the interest rate and the interest is compounded continuously. Assume that no withdrawals or further deposits are made. Initial amount: \(\$ 3000 ; r=4 \%\)

In Exercises \(53-64,\) simplify each expression. Assume that each variable expression is defined for appropriate values of \(x .\) Do not use a calculator. $$\log _{5} 625$$

For an initial deposit of \(\$ 1500\) , find the total amount in a bank account after 5 years for the interest rates and compounding frequencies given. 3% compounded semiannually

Find the domain of each function. Use your answer to help you graph the function, and label all asymptotes. $$h(x)=\ln x+2$$

Evaluate the given quantity by referring to the function \(f\) given in the following table. $$\begin{array}{cc}x & f(x) \\\\-2 & 1 \\\\-1 & 2 \\\0 & 0 \\\1 & -1 \\\2 & -2\end{array}$$ $$f^{-1}(1)$$

Determine how long it takes for the given investment to double if \(r\) is the interest rate and the interest is compounded continuously. Assume that no withdrawals or further deposits are made. Initial amount: \(\$ 4000 ; r=5.75 \%\)

In Exercises \(53-64,\) simplify each expression. Assume that each variable expression is defined for appropriate values of \(x .\) Do not use a calculator. $$\log _{a} \sqrt[5]{a^{2}}, a > 0, a \neq 1$$

For an initial deposit of \(\$ 1500\) , find the total amount in a bank account after 5 years for the interest rates and compounding frequencies given. 6% compounded monthly

Find the domain of each function. Use your answer to help you graph the function, and label all asymptotes. $$f(x)=\log _{4}(x+1)$$

Evaluate the given quantity by referring to the function \(f\) given in the following table. $$\begin{array}{cc}x & f(x) \\\\-2 & 1 \\\\-1 & 2 \\\0 & 0 \\\1 & -1 \\\2 & -2\end{array}$$ $$f^{-1}(2)$$

Determine how long it takes for the given investment to double if \(r\) is the interest rate and the interest is compounded continuously. Assume that no withdrawals or further deposits are made. Initial amount: \(\$ 6000 ; r=6.25 \%\)

In Exercises \(53-64,\) simplify each expression. Assume that each variable expression is defined for appropriate values of \(x .\) Do not use a calculator. $$\log _{b} \sqrt[3]{b}, b > 0, b \neq 1$$

For an initial deposit of \(\$ 1500\) , find the total amount in a bank account after 5 years for the interest rates and compounding frequencies given. 3% compounded quarterly

Find the domain of each function. Use your answer to help you graph the function, and label all asymptotes. $$f(x)=\log _{5}(x-2)$$

Evaluate the given quantity by referring to the function \(f\) given in the following table. $$\begin{array}{cc}x & f(x) \\\\-2 & 1 \\\\-1 & 2 \\\0 & 0 \\\1 & -1 \\\2 & -2\end{array}$$ $$f^{-1}\left(f^{-1}(-2)\right)$$

Determine how long it takes for the given investment to double if \(r\) is the interest rate and the interest is compounded continuously. Assume that no withdrawals or further deposits are made. Initial amount: \(\$ 2700 ; r=7.5 \%\)

In Exercises \(65-68,\) use a graphing utility with a decimal window. Graph \(f(x)=\log 10 x\) and \(g(x)=\log x\) on the same set of axes. Explain the relationship between the two graphs in terms of the properties of logarithms.

For an initial deposit of \(\$ 1500,\) find the total amount in a bank account after \(t\) years for the interest rates and values of t given, assuming continuous compounding of interest. \(6 \%\) interest; \(t=3\)

Find the domain of each function. Use your answer to help you graph the function, and label all asymptotes. $$f(x)=\ln (x+4)$$

Determine how long it takes for the given investment to double if \(r\) is the interest rate and the interest is compounded continuously. Assume that no withdrawals or further deposits are made. Initial amount: \(\$ 3800 ; r=5.8 \%\)

In Exercises \(65-68,\) use a graphing utility with a decimal window. Graph \(f(x)=\log 0.1 x\) and \(g(x)=\log x\) on the same set of axes. Explain the relationship between the two graphs in terms of the properties of logarithms.

For an initial deposit of \(\$ 1500,\) find the total amount in a bank account after \(t\) years for the interest rates and values of t given, assuming continuous compounding of interest. \(7 \%\) interest; \(t=4\)

Find the domain of each function. Use your answer to help you graph the function, and label all asymptotes. $$f(x)=\log (x-3)$$

In Exercises \(65-68,\) use a graphing utility with a decimal window. Graph \(f(x)=\ln e^{2} x\) and \(g(x)=\ln x\) on the same set of coordinate axes. Explain the relationship between the two graphs in terms of the properties of logarithms.

For an initial deposit of \(\$ 1500,\) find the total amount in a bank account after \(t\) years for the interest rates and values of t given, assuming continuous compounding of interest. \(3.25 \%\) interest; \(t=5.5\)

Find the domain of each function. Use your answer to help you graph the function, and label all asymptotes. $$g(x)=2 \log _{3}(x-1)$$

Find the interest rate \(r\) if the interest on the initial deposit is compounded continuously and no withdrawals or further deposits are made. Initial amount: \(\$ 3000 ;\) Amount in 3 years: \(\$ 3600\)

In Exercises \(65-68,\) use a graphing utility with a decimal window. Graph \(f(x)=\log x-\log (x-1)\) and \(g(x)=\log \frac{x}{x-1}\) on the same set of axes. (a) What are the domains of the two functions? (b) For what values of \(x\) do these two functions agree? (c) To what extent does this pair of functions exhibit the quotient property of logarithms?

For an initial deposit of \(\$ 1500,\) find the total amount in a bank account after \(t\) years for the interest rates and values of t given, assuming continuous compounding of interest. \(4.75 \%\) interest; \(t=6.5\)

Find the domain of each function. Use your answer to help you graph the function, and label all asymptotes. $$f(x)=-\log _{2}(x+3)$$

Find the interest rate \(r\) if the interest on the initial deposit is compounded continuously and no withdrawals or further deposits are made. Initial amount: \(\$ 4000 ;\) Amount in 8 years: \(\$ 6000\)

Fill in the table according to the given rule and find an expression for the function represented by the rule. Salary The annual salary of an employee at a certain company starts at \(\$ 10,000\) and is increased by \(5 \%\) at the end of every year. $$\begin{aligned} &\begin{array}{cc} \text { Years at } & \text { Annual } \\ \text { Work } & \text { Salary } \end{array}\\\ &\begin{array}{l} 0 \\ 1 \\ 2 \\ 3 \\ 4 \end{array} \end{aligned}$$

Refer to the definition of pH in Example 5 to solve Exercises \(69-73\). Suppose solution A has a pH of 5 and solution B has a pH of \(9 .\) What is the ratio of the concentration of hydrogen ions in solution \(A\) to the concentration of hydrogen ions in solution B?

Find the domain of each function. Use your answer to help you graph the function, and label all asymptotes. $$f(t)=\log _{1 / 3} t$$

Find the interest rate \(r\) if the interest on the initial deposit is compounded continuously and no withdrawals or further deposits are made. Initial amount: \(\$ 6000 ;\) Amount in 10 years: \(\$ 12,000\)

Refer to the definition of pH in Example 5 to solve Exercises \(69-73\). Find the \(\mathrm{pH}\) of a solution with \(\left[\mathrm{H}^{+}\right]=4 \times 10^{-5}\).

Fill in the table according to the given rule and find an expression for the function represented by the rule. A population of cockroaches starts out at 100 and doubles every month. $$\begin{aligned} &\begin{array}{cc} \text { Month } & \text { Population } \end{array}\\\ &\begin{array}{l} 0 \\ 1 \\ 2 \\ 3 \\ 4 \end{array} \end{aligned}$$

Find the domain of each function. Use your answer to help you graph the function, and label all asymptotes. $$g(s)=\log _{1 / 2} s$$

Applications In this set of exercises, you will use inverse functions to study real-world problems. Find a function that converts \(x\) gallons into quarts. Find its inverse and explain what it does.

Find the interest rate \(r\) if the interest on the initial deposit is compounded continuously and no withdrawals or further deposits are made. Initial amount: $$ 8500 ;\( Amount in 5 years: $$ 10,000\)

Refer to the definition of pH in Example 5 to solve Exercises \(69-73\). Find the \(\mathrm{pH}\) of a solution with \(\left[\mathrm{H}^{+}\right]=6 \times 10^{-8}\).