Chapter 5

Simplify each expression. (a) \(3^{\log _{3} 7}\) (b) \(4^{\log _{4} 9}\) (c) \(12^{\log _{13} 4}\) (d) \(a^{\log _{c} k}(k > 0, a > 0, a \neq 1)\)

If \(f(x)=x,\) then for any function \(g,(f \circ g)(x)=\) \((g \circ f)(x)=\)______.

Solve each exponential equation. Express the solution set so that (a) solutions are in exact form and, if irrational, (b) solutions are approximated to the nearest thousandth. Support your solutions by using a calculator. $$30-3(0.75)^{x-1}=29$$

Simplify each expression. (a) \(\log _{3} 3^{19}\) (b) \(\log _{4} 4^{17}\) (c) \(\log _{12} 12^{1 / 3}\) (d) \(\log _{a} \sqrt{a}(a>0, a \neq 1)\)

The interest rate stated by a financial institution is sometimes called the nominal rate. If interest is compounded, the actual rate is, in general, higher than the nominal rate, and is called the effective rate. If \(r\) is the nominal rate and \(n\) is the number of times interest is compounded annually, then $$R=\left(1+\frac{r}{n}\right)^{n}-1$$ is the effective rate. Here, \(R\) represents the annual rate that the investment would earn if simple interest were paid. Find the effective rate to the nearest hundredth of a percent if the nominal rate is \(3 \%\) and interest is compounded quarterly.

If a function \(f\) has an inverse, then the graph of \(f^{-1}\) may be obtained by reflecting the graph of \(f\) across the line with equation ____.

Solve each logarithmic equation. Express all solutions in exact form. Support your solutions by using a calculator. $$5 \ln x=10$$

Simplify each expression. (a) \(\log _{3} 1\) (b) \(\log _{4} 4^{17}\) (c) \(\log _{12} 1\) (d) \(\log _{a} 1(a>0, a \neq 1)\)

The interest rate stated by a financial institution is sometimes called the nominal rate. If interest is compounded, the actual rate is, in general, higher than the nominal rate, and is called the effective rate. If \(r\) is the nominal rate and \(n\) is the number of times interest is compounded annually, then $$R=\left(1+\frac{r}{n}\right)^{n}-1$$ is the effective rate. Here, \(R\) represents the annual rate that the investment would earn if simple interest were paid. Estimate the effective rate if the nominal rate is \(4.5 \%\) and interest is compounded daily \((n=365)\)

If a function \(f\) has an inverse and \(f(-3)=6,\) then \(f^{-1}(6)=\)____.

Solve each logarithmic equation. Express all solutions in exact form. Support your solutions by using a calculator. $$3 \log x=2$$

Simplify each expression. (a) Explain in your own words the meaning of \(\log _{a} x\) (b) In the expression \(\log _{a} x,\) why must \(x\) be nonnegative?

Sketch the graph of \(f(x)=2^{x}\). Then refer to it and use the techniques of Chapter 2 to graph each function. $$f(x)=2^{x}-4$$

In the formula \(A=P\left(1+\frac{r}{n}\right)^{n t},\) we can interpret \(P\) as the present value of A dollars t years from now, earning annual interest \(r\) compounded \(n\) times per year. In this context, \(A\) is called the future value. If we solve the formula for \(P,\) we obtain $$P=A\left(1+\frac{r}{n}\right)^{-n t}$$ Use the future value formula. Find the present value of an account that will be worth \(\$ 10,000\) in 5 years, if interest is compounded semiannually at \(3 \%\).

Solve each logarithmic equation. Express all solutions in exact form. Support your solutions by using a calculator. $$\ln (4 x)=1.5$$

Evaluate each expression. Do not use a calculator. $$\log 10^{1.5}$$

In the formula \(A=P\left(1+\frac{r}{n}\right)^{n t},\) we can interpret \(P\) as the present value of A dollars t years from now, earning annual interest \(r\) compounded \(n\) times per year. In this context, \(A\) is called the future value. If we solve the formula for \(P,\) we obtain $$P=A\left(1+\frac{r}{n}\right)^{-n t}$$ Use the future value formula. Find the present value of an account that will be worth \(\$ 25,000\) in 2.75 years, if interest is compounded quarterly at \(6 \%\).

If \(f\) is a function that has an inverse and the graph of \(f\) lies completely within the second quadrant, then the graph of \(f^{-1}\) lies completely within the ____ quadrant.

Solve each logarithmic equation. Express all solutions in exact form. Support your solutions by using a calculator. $$\ln (2 x)=5$$

Evaluate each expression. Do not use a calculator. $$\log 10^{4.3}$$

In the formula \(A=P\left(1+\frac{r}{n}\right)^{n t},\) we can interpret \(P\) as the present value of A dollars t years from now, earning annual interest \(r\) compounded \(n\) times per year. In this context, \(A\) is called the future value. If we solve the formula for \(P,\) we obtain $$P=A\left(1+\frac{r}{n}\right)^{-n t}$$ Use the future value formula. Estimate the interest rate necessary for a present value of \(\$ 25,000\) to grow to a future value of \(\$ 30,416\) if interest is compounded annually for 5 years.

Use the definition of inverse functions to show analytically that \(f\) and \(g\) are inverses. $$f(x)=3 x-7, \quad g(x)=\frac{x+7}{3}$$

Solve each logarithmic equation. Express all solutions in exact form. Support your solutions by using a calculator. $$\log (2-x)=0.5$$

Evaluate each expression. Do not use a calculator. $$\log 10^{\sqrt{5}}$$

In the formula \(A=P\left(1+\frac{r}{n}\right)^{n t},\) we can interpret \(P\) as the present value of A dollars t years from now, earning annual interest \(r\) compounded \(n\) times per year. In this context, \(A\) is called the future value. If we solve the formula for \(P,\) we obtain $$P=A\left(1+\frac{r}{n}\right)^{-n t}$$ Use the future value formula. Estimate the interest rate necessary for a present value of \(\$ 1200\) to grow to a future value of \(\$ 1408\) if interest is compounded quarterly for 8 years.

Use the definition of inverse functions to show analytically that \(f\) and \(g\) are inverses. $$f(x)=4 x+3, \quad g(x)=\frac{x-3}{4}$$

Solve each logarithmic equation. Express all solutions in exact form. Support your solutions by using a calculator. $$\ln (1-x)=\frac{1}{2}$$

Evaluate each expression. Do not use a calculator. $$\log 10^{\sqrt{3}}$$

Graph each function. $$f(x)=\log _{5} x$$

Use the definition of inverse functions to show analytically that \(f\) and \(g\) are inverses. $$f(x)=x^{3}+4, \quad g(x)=\sqrt[3]{x-4}$$

Solve each logarithmic equation. Express all solutions in exact form. Support your solutions by using a calculator. $$\log _{6}(2 x+4)=2$$

Evaluate each expression. Do not use a calculator. $$\ln e^{2 / 3}$$

Use the amortization formulas given in this section to find (a) the monthly payment on a loan with the given conditions and (b) the total interest that will be paid during the term of the loan. \(\$ 8500\) is amortized over 4 years with an interest rate of \(7.5 \%\).

Use the amortization formulas given in this section to find (a) the monthly payment on a loan with the given conditions and (b) the total interest that will be paid during the term of the loan. \(\$ 9600\) is amortized over 5 years with an interest rate of \(5.2 \%\)

Graph each function. $$f(x)=\log _{10} x$$

Use the definition of inverse functions to show analytically that \(f\) and \(g\) are inverses. $$f(x)=x^{3}-7, \quad g(x)=\sqrt[3]{x+7}$$

Solve each logarithmic equation. Express all solutions in exact form. Support your solutions by using a calculator. $$\log _{5}(8-3 x)=3$$

Evaluate each expression. Do not use a calculator. $$\ln e^{0.5}$$

Graph each function. $$f(x)=\log _{1 / 2}(1-x)$$

Solve each equation. $$2^{3-x}=8$$

Use the definition of inverse functions to show analytically that \(f\) and \(g\) are inverses. $$f(x)=-x^{5}, \quad g(x)=-\sqrt[5]{x}$$

Solve each logarithmic equation. Express all solutions in exact form. Support your solutions by using a calculator. $$\log _{4}\left(x^{3}+37\right)=3$$

Evaluate each expression. Do not use a calculator. $$\ln e^{\pi}$$

Use the amortization formulas given in this section to find (a) the monthly payment on a loan with the given conditions and (b) the total interest that will be paid during the term of the loan. \(\$ 125,000\) is amortized over 30 years with an interest rate of \(7.25 \%\)

Graph each function. $$f(x)=\log _{1 / 3}(3-x)$$

Solve each equation. $$5^{2 x+1}=25$$

Solve each logarithmic equation. Express all solutions in exact form. Support your solutions by using a calculator. $$\log _{7}\left(x^{3}+65\right)=0$$

Evaluate each expression. Do not use a calculator. $$\ln e^{\sqrt{6}}$$

Use the table feature of your graphing calculator to work parts (a) and (b). (a) Find how long it will take \(\$ 1500\) invested at \(5.75 \%\) compounded daily, to triple in value. Locate the solution by systematically decreasing \(\Delta\) Tbl. Find the answer to the nearest day. (Find your answer to the nearest day by eventually letting \(\Delta \mathrm{Tbl}=\frac{1}{365} .\) The decimal part of the solution can be multiplied by 365 to determine the number of days greater than the nearest year. For example, if the solution is determined to be 16.2027 years, then multiply 0.2027 by 365 to get \(73.9855 .\) The solution is then, to the nearest day, 16 years and 74 days.) Confirm your answer analytically. (b) Find how long it will take \(\$ 2000\) invested at \(8 \%,\) compounded daily, to be worth \(\$ 5000\).

Graph each function. $$f(x)=\log _{3}(x-1)$$