Chapter 5

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. $$e^{x-3}=2^{3 x}$$

Decide whether each function is one-to-one. Do not use a calculator. $$y=-\sqrt[3]{x+5}$$

Solve each equation. Give the exact answer. $$\log _{x} 3^{12}=24$$

Suppose that \(\$ 2500\) is invested in an account that pays interest compounded continuously. Find the amount of time that it would take for the account to grow to the given amount at the given rate of interest. Round to the nearest tenth of a year. \(\$ 3500\) at \(4.25 \%\)

Graph each function by hand and support your sketch with a calculator graph. Give the domain, range, and equation of the asymptote. Determine if \(f\) is increasing or decreasing on its domain. $$f(x)=\left(\frac{2}{3}\right)^{x}$$

Find the domain of each logarithmic function analytically. You may wish to support your answer graphically. $$y=\log \left(\frac{x+1}{x-5}\right)$$

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. $$e^{0.5 x}=3^{1-2 x}$$

Decide whether each function is one-to-one. Do not use a calculator. $$y=\frac{1}{x+2}$$

Solve each equation. Give the exact answer. $$\log _{x} 5^{2}=6$$

Suppose that \(\$ 2500\) is invested in an account that pays interest compounded continuously. Find the amount of time that it would take for the account to grow to the given amount at the given rate of interest. Round to the nearest tenth of a year. \(\$ 5000\) at \(5 \%\)

Graph each function by hand and support your sketch with a calculator graph. Give the domain, range, and equation of the asymptote. Determine if \(f\) is increasing or decreasing on its domain. $$f(x)=e^{x}$$

Find the domain of each logarithmic function analytically. You may wish to support your answer graphically. $$y=\log |3 x-7|$$

Decide whether each function is one-to-one. Do not use a calculator. $$y=\frac{-4}{x-8}$$

Solve each equation. Give the exact answer. $$\log _{6} x=-3$$

Suppose that \(\$ 2500\) is invested in an account that pays interest compounded continuously. Find the amount of time that it would take for the account to grow to the given amount at the given rate of interest. Round to the nearest tenth of a year. \(\$ 5000\) at \(6 \%\)

Graph each function by hand and support your sketch with a calculator graph. Give the domain, range, and equation of the asymptote. Determine if \(f\) is increasing or decreasing on its domain. $$f(x)=-e^{x}$$

Find the domain of each logarithmic function analytically. You may wish to support your answer graphically. $$y=\log |6 x+6|$$

Decide whether each function is one-to-one. Do not use a calculator. $$f(x)=-7$$

Solve each equation. Give the exact answer. $$\log _{4} x=-\frac{1}{6}$$

The time \(T\) in years it takes for a principal of \(\$ 1000\) receiving \(2 \%\) annual interest compounded continuously to reach an amount \(A\) is calculated by the following logarithmic function. $$T(A)=50 \ln \frac{A}{1000}$$ (a) Find a reasonable domain for \(T\). Interpret your answer. (b) How many years does it take the principal to grow to \(\$ 1200 ?\) (c) Determine the amount in the account after 23.5 years by solving the equation \(T(A)=23.5\)

Graph each function by hand and support your sketch with a calculator graph. Give the domain, range, and equation of the asymptote. Determine if \(f\) is increasing or decreasing on its domain. $$f(x)=e^{x+1}$$

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. $$0.05(1.15)^{x}=5$$

Decide whether each function is one-to-one. Do not use a calculator. $$f(x)=\left\\{\begin{array}{r}3 \text { if } x \geq 0 \\\\-x \text { if } x<0\end{array}\right.$$

Solve each equation. Give the exact answer. $$\log _{x} 16=\frac{4}{3}$$

Suppose that a sample of bacteria has a concentration of 2 million bacteria per milliliter and it doubles in concentration every 12 hours. Then the time \(T\) it takes for the sample to reach a concentration of \(C\) can be approximated by the following logarithmic function. \(T(C)=\frac{500}{29} \ln \frac{C}{2}\) (a) Find the domain of \(T .\) Interpret your answer. (b) How long does it take for the concentration of bacteria to increase by \(50 \% ?\) (c) Determine the concentration \(C\) after 15 hours by solving the equation \(T(C)=15\)

Graph each function by hand and support your sketch with a calculator graph. Give the domain, range, and equation of the asymptote. Determine if \(f\) is increasing or decreasing on its domain. $$f(x)=e^{x}-1$$

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. $$1.2(0.9)^{x}=0.6$$

(a) Explain why a polynomial function of even degree with domain \((-\infty, \infty)\) cannot be one-to-one. (b) Explain why in some cases a polynomial function of odd degree with domain \((-\infty, \infty)\) is not one-to-one.

Solve each equation. Give the exact answer. $$\log _{x} 81=4$$

Account Estimate how long it will take for \(\$ 1000\) to grow to \(\$ 5000\) at an interest rate of \(3.5 \%\) if interest is compounded (a) quarterly; (b) continuously.

Graph each function by hand and support your sketch with a calculator graph. Give the domain, range, and equation of the asymptote. Determine if \(f\) is increasing or decreasing on its domain. $$f(x)=10^{x}$$

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. $$3(2)^{x-2}+1=100$$

Solve each equation. Give the exact answer. $$\log _{2}(x+1)=3$$

Account Estimate how long it will take for \(\$ 5000\) to grow to \(\$ 8400\) at an interest rate of \(6 \%\) if interest is compounded (a) semiannually; (b) continuously.

Graph each function by hand and support your sketch with a calculator graph. Give the domain, range, and equation of the asymptote. Determine if \(f\) is increasing or decreasing on its domain. $$f(x)=10^{-x}$$

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. $$5(1.2)^{3 x-2}+1=11$$

For \(f\) to be one-to-one, if \(a \neq b,\) then ____.

Solve each equation. Give the exact answer. $$\log _{3}(x-1)=2$$

Tom Tupper wants to buy a \(\$ 30,000\) car. He has saved \(\$ 27,000 .\) Find the number of years (to the nearest tenth) it will take for his \(\$ 27,000\) to grow to \(\$ 30,000\) at \(2.3 \%\) interest compounded quarterly.

Graph each function by hand and support your sketch with a calculator graph. Give the domain, range, and equation of the asymptote. Determine if \(f\) is increasing or decreasing on its domain. $$f(x)=4^{-x}$$

If \(f\) and \(g\) are inverses, then \((f \circ g)(x)=\) ____ and ____=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. $$2(1.05)^{x}+3=10$$

Solve each equation. Give the exact answer. $$\log _{9} \frac{\sqrt[4]{27}}{3}=x$$

Estimate the doubling time of an investment earning \(2.5 \%\) interest if interest is compounded (a) quarterly; (b) continuously.

Graph each function by hand and support your sketch with a calculator graph. Give the domain, range, and equation of the asymptote. Determine if \(f\) is increasing or decreasing on its domain. $$f(x)=6^{-x}$$

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. $$3(1.4)^{x}-4=60$$

Solve each equation. Give the exact answer. $$\log _{1 / 4} \frac{16^{2}}{2^{-3}}=x$$

A construction worker wants to invest \(\$ 60,000\) in a pension plan. One investment offers \(2 \%\) compounded quarterly. Another offers \(1.8 \%\) compounded continuously. Which investment will earn more interest in 5 years? How much more will the better plan earn?

If the point \((a, b)\) lies on the graph of \(f\) and \(f\) has an inverse, then the point ____ lies on the graph of \(f^{-1}\).

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. $$5(1.015)^{x-1980}=8$$