Chapter 5

Solve each logarithmic equation. Express all solutions in exact form. Support your solutions by using a calculator. $$\log _{2}\left(\log _{2} x\right)=1$$

Find the hydronium ion \(\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]\) concentration for each substance with the given \(p H\) Soda pop, 2.7

Solve each equation in part (a) analytically. Support your answer with a calculator graph. Then use the graph to solve the associated inequalities in parts (b) and (c). (a) \(\left(\frac{1}{2}\right)^{-x}=\left(\frac{1}{4}\right)^{x+1}\) (b) \(\left(\frac{1}{2}\right)^{-x} \geq\left(\frac{1}{4}\right)^{x+1}\) (c) \(\left(\frac{1}{2}\right)^{-x} \leq\left(\frac{1}{4}\right)^{x+1}\)

For each function that is one-to-one, write an equation for the inverse function in the form \(y=f^{-1}(x)\) and then graph \(f\) and \(f^{-1}\) on the same axes. Give the domain and range of \(f\) and \(f^{-1}\). If the function is not one-to-one, say so. $$y=\frac{1}{x}$$

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

Use a graphing calculator to solve each equation. Give solutions to the nearest hundredth. \(\log _{10} x=x-2\)

Find the hydronium ion \(\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]\) concentration for each substance with the given \(p H\) Wine, 3.4

Solve each equation in part (a) analytically. Support your answer with a calculator graph. Then use the graph to solve the associated inequalities in parts (b) and (c). (a) \(\left(\frac{2}{3}\right)^{x-1}=\left(\frac{81}{16}\right)^{x+1}\) (b) \(\left(\frac{2}{3}\right)^{x-1} \leq\left(\frac{81}{16}\right)^{x+1}\) (c) \(\left(\frac{2}{3}\right)^{x-1} \geq\left(\frac{81}{16}\right)^{x+1}\)

For each function that is one-to-one, write an equation for the inverse function in the form \(y=f^{-1}(x)\) and then graph \(f\) and \(f^{-1}\) on the same axes. Give the domain and range of \(f\) and \(f^{-1}\). If the function is not one-to-one, say so. $$y=\frac{4}{x}$$

Solve each logarithmic equation. Express all solutions in exact form. Support your solutions by using a calculator. $$\ln (\ln x)=0$$

Use a graphing calculator to solve each equation. Give solutions to the nearest hundredth. $$2^{-x}=\log _{10} x$$

Find the hydronium ion \(\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]\) concentration for each substance with the given \(p H\) Beer, 4.8

For each function that is one-to-one, write an equation for the inverse function in the form \(y=f^{-1}(x)\) and then graph \(f\) and \(f^{-1}\) on the same axes. Give the domain and range of \(f\) and \(f^{-1}\). If the function is not one-to-one, say so. $$y=\frac{2}{x+3}$$

Concept Check If \(f(x)=a^{x}\) and \(f(3)=27,\) find each value of \(f(x)\) (a) \(f(1)\) (b) \(f(-1)\) (c) \(f(2)\) (d) \(f(0)\)

Use a graphing calculator to solve each equation. Give solutions to the nearest hundredth. $$e^{x}=x^{2}$$

Give an exponential function in the form \(f(x)=a^{x}\) whose graph contains the given point. \((3,8)\)

Find the hydronium ion \(\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]\) concentration for each substance with the given \(p H\) Drinking water, 6.5

For each function that is one-to-one, write an equation for the inverse function in the form \(y=f^{-1}(x)\) and then graph \(f\) and \(f^{-1}\) on the same axes. Give the domain and range of \(f\) and \(f^{-1}\). If the function is not one-to-one, say so. $$y=\frac{3}{x-4}$$

Use the properties of logarithms to rewrite each logarithm if possible. Assume that all variables represent positive real numbers. $$\log _{3} \frac{2}{5}$$

For each function that is one-to-one, write an equation for the inverse function in the form \(y=f^{-1}(x)\) and then graph \(f\) and \(f^{-1}\) on the same axes. Give the domain and range of \(f\) and \(f^{-1}\). If the function is not one-to-one, say so. $$f(x)=\sqrt{6+x}, x \geq-6$$

A student told a friend, "You must reject any negative solution of an equation involving logarithms." Is this correct? Write an explanation of your answer.

Use properties of exponents to write each function in the form \(f(t)=k a^{\prime},\) where \(k\) is a constant. (Hint: Recall that \(a^{x+y}=a^{x} \cdot a^{y}\).) $$f(t)=3^{2 t+3}$$

Use the properties of logarithms to rewrite each logarithm if possible. Assume that all variables represent positive real numbers. $$\log _{4} \frac{6}{7}$$

For each function that is one-to-one, write an equation for the inverse function in the form \(y=f^{-1}(x)\) and then graph \(f\) and \(f^{-1}\) on the same axes. Give the domain and range of \(f\) and \(f^{-1}\). If the function is not one-to-one, say so. $$f(x)=-\sqrt{x^{2}-16}, x \geq 4$$

The age in years of a female blue whale is approximated by $$=-2.57 \ln \left(\frac{87-L}{63}\right)$$ where \(L\) is its length in feet. (a) How old is a female blue whale that measures 80 feet? (b) Estimate the length of a female blue whale that is 4 years old. (c) The equation that defines t has domain 24

Barometric Pressure The function $$f(x)=27+1.105 \log (x+1)$$ approximates the barometric pressure in inches of mercury at a distance of x miles from the eye of a hurricane. (Source: Miller, A. and R. Anthes, Meteorology, Fifth Edition, Charles E. Merrill.) (a) Approximate the pressure 9 miles from the eye of the hurricane. (b) The ordered pair (99,29.21) belongs to this function. What information does it convey?

Use properties of exponents to write each function in the form \(f(t)=k a^{\prime},\) where \(k\) is a constant. (Hint: Recall that \(a^{x+y}=a^{x} \cdot a^{y}\).) $$f(t)=\left(\frac{1}{3}\right)^{1-2 t}$$

Use the properties of logarithms to rewrite each logarithm if possible. Assume that all variables represent positive real numbers. $$\log _{2} \frac{6 x}{y}$$

The given function \(f\) is one-to-one. Find \(f^{-1}(x)\). $$f(x)=\frac{4 x}{x+1}$$

Use a graphing calculator to find the solution of each equation. Round your result to the nearest thousandth. $$1.5^{\log x}=e^{0.5}$$

Sprinter's Speed and Time During the 100 -meter dash, the elapsed time \(T\) for a sprinter to reach a speed of x meters per second is given by the following function. $$T(x)=-1.2 \ln \left(1-\frac{x}{11}\right)$$ (a) How much time had elapsed when the sprinter was running 0 meters per second? Interpret your answer. (b) At the end of the race, the sprinter was moving at 10.998 meters per second. What was the sprinter's time for this 100 -meter dash? (c) Find T^{-1}(x) and interpret its meaning.

The graph of \(y=e^{x-3}\) can be obtained by translating the graph of \(y=e^{x}\) to the right 3 units. Find a constant \(C\) such that the graph of \(y=C e^{x}\) is the same as the graph of \(y=e^{x-3} .\) Verify your result by graphing both functions.

Use the properties of logarithms to rewrite each logarithm if possible. Assume that all variables represent positive real numbers. $$\log _{3} \frac{4 p}{q}$$

The given function \(f\) is one-to-one. Find \(f^{-1}(x)\). $$f(x)=\frac{3 x}{5-x}$$

Use a graphing calculator to find the solution of each equation. Round your result to the nearest thousandth. $$1.5^{\ln x}=10^{0.5}$$

Use the appropriate compound interest formula to find the amount that will be in each account, given the stated conditions. \(\$ 20,000\) invested at \(3 \%\) annual interest for 4 years compounded (a) annually; (b) semiannually

Use the properties of logarithms to rewrite each logarithm if possible. Assume that all variables represent positive real numbers. $$\log _{5} \frac{5 \sqrt{7}}{3 m}$$

The given function \(f\) is one-to-one. Find \(f^{-1}(x)\). $$f(x)=\frac{1-2 x}{3 x}$$

$$\text { Solve each formula for the indicated variable.}$$ $$r=p-k \ln t, \text { for } t$$

Use the appropriate compound interest formula to find the amount that will be in each account, given the stated conditions. \(\$ 35,000\) invested at \(4.2 \%\) annual interest for 3 years compounded (a) annually; (b) quarterly

Use the properties of logarithms to rewrite each logarithm if possible. Assume that all variables represent positive real numbers. $$\log _{2} \frac{2 \sqrt{3}}{5 p}$$

The given function \(f\) is one-to-one. Find \(f^{-1}(x)\). $$f(x)=\frac{4-x}{5 x}$$

$$\text { Solve each formula for the indicated variable.}$$ $$p=a+\frac{k}{\ln x}, \text { for } x$$

Use the appropriate compound interest formula to find the amount that will be in each account, given the stated conditions. \(\$ 27,500\) invested at \(3.95 \%\) annual interest for 5 years compounded (a) daily \((n=365) ;\) (b) continuously

Use the properties of logarithms to rewrite each logarithm if possible. Assume that all variables represent positive real numbers. $$\log _{4}(2 x+5 y)$$

The given function \(f\) is one-to-one. Find \(f^{-1}(x)\). $$f(x)=\sqrt{x^{2}-4}, x \geq 2$$

$$\text { Solve each formula for the indicated variable.}$$ $$T=T_{0}+\left(T_{1}-T_{0}\right) 10^{-k_{1}}, \text { for } t$$

Use the appropriate compound interest formula to find the amount that will be in each account, given the stated conditions. \(\$ 15,800\) invested at \(1.6 \%\) annual interest for 6.5 years compounded (a) quarterly; (b) continuously

Use the properties of logarithms to rewrite each logarithm if possible. Assume that all variables represent positive real numbers. $$\log _{6}(7 m+3 q)$$

The given function \(f\) is one-to-one. Find \(f^{-1}(x)\). $$f(x)=\sqrt{x-8}, x \geq 8$$