Chapter 5

Solve each equation. $$12^{x-3}=1$$

Determine whether each function is one-to-one. If so, find its inverse. $$f=\\{(10,4),(20,5),(30,6),(40,7)\\}$$

Solve each logarithmic equation. Express all solutions in exact form. Support your solutions by using a calculator. $$\ln x+\ln x^{2}=3$$

Evaluate each expression. Do not use a calculator. $$\sqrt{7} \ln e^{\sqrt{7}}$$

If interest is compounded continuously and the interest rate is tripled, what effect will this have on the time required for an investment to double?

Graph each function. $$f(x)=\log _{2}\left(x^{2}\right)$$

Solve each equation. $$3^{5-x}=1$$

Determine whether each function is one-to-one. If so, find its inverse. $$g=\\{(5,12),(10,22),(15,32),(20,42)\\}$$

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

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

Escherichia coli is a strain of bacteria that occurs naturally in many organisms. Under certain conditions, the number of bacteria present in a colony is approximated by $$A(t)=A_{0} e^{0.023 t}$$ where \(t\) is in minutes. If \(A_{0}=2,400,000,\) find the number of bacteria at each time. Round to the nearest hundred thousand. (a) 5 minutes (b) 10 minutes (c) 60 minutes

Explain how the graph of the given function can be obtained from the graph of y=\log _{2} x, and (b) graph the function. $$y=\log _{2}(x+4)$$

Solve each equation. $$e^{4 x-1}=\left(e^{2}\right)^{x}$$

Determine whether each function is one-to-one. If so, find its inverse. $$f=\\{(1,5),(2,6),(3,5),(4,8)\\}$$

Solve each logarithmic equation. Express all solutions in exact form. Support your solutions by using a calculator. $$2 \ln (x-1)+30=34$$

Use a calculator to find a decimal approximation for each common or natural logarithm. $$\log 43$$

The growth of bacteria in food products makes it necessary to date some products (such as milk) so that they will be sold and consumed before the bacterial count becomes too high. Suppose that, under certain storage conditions, the number of bacteria present in a product is $$f(t)=500 e^{0.1 t}$$ where \(t\) is time in days after packing of the product and the value of \(f(t)\) is in millions. (a) If the product cannot be safely eaten after the bacterial count reaches \(3,000,000,000,\) how long will this take? (b) If \(t=0\) corresponds to January \(1,\) what date should be placed on the product?

Explain how the graph of the given function can be obtained from the graph of y=\log _{2} x, and (b) graph the function. $$y=\log _{2}(x-6)$$

Solve each equation. $$e^{3-x}=\left(e^{3}\right)^{-x}$$

Use a calculator to find a decimal approximation for each common or natural logarithm. $$\log 1247$$

Solve each logarithmic equation. Express all solutions in exact form. Support your solutions by using a calculator. $$1-4 \ln (2 x-1)=-5$$

Determine whether each function is one-to-one. If so, find its inverse. $$g=\\{(0,10),(1,20),(2,10),(3,40)\\}$$

The revenue in millions of dollars for the first 5 years of mobile advertising is given by \(A(x)=42(2)^{x},\) where \(x\) is years after the industry started. (Source: Business Insider.) (a) Determine analytically when revenue was about \(\$ 400\) million. (b) Solve part (a) graphically. (c) According to this model, when did the mobile advertising revenue reach \(\$ 1\) billion?

Explain how the graph of the given function can be obtained from the graph of y=\log _{2} x, and (b) graph the function. $$y=3 \log _{2} x+1$$

Solve each equation. $$27^{4 x}=9^{x+1}$$

Use a calculator to find a decimal approximation for each common or natural logarithm. $$\log 0.783$$

Determine whether each function is one-to-one. If so, find its inverse. $$f=\left\\{\left(0,0^{2}\right),\left(1,1^{2}\right),\left(2,2^{2}\right),\left(3,3^{2}\right),\left(4,4^{2}\right)\right\\}$$

Solve each logarithmic equation. Express all solutions in exact form. Support your solutions by using a calculator. $$5 \log \left(x^{2}-1\right)+7=12$$

The revenue in millions of dollars for the first 5 years of Internet advertising is given by \(A(x)=25(2.95)^{x},\) where \(x\) is years after the industry started. (Source: Business Insider.) (a) What was the Internet advertising revenue after 5 years? (b) Determine analytically when revenue was about \(\$ 250\) million. (c) According to this model, when did the Internet advertising revenue reach \(\$ 1\) billion?

Explain how the graph of the given function can be obtained from the graph of y=\log _{2} x, and (b) graph the function. $$y=-4 \log _{2} x-8$$

Solve each equation. $$32^{x}=16^{1-x}$$

Use a calculator to find a decimal approximation for each common or natural logarithm. $$\log 0.014$$

Determine whether each function is one-to-one. If so, find its inverse. $$g=\left\\{\left(0,0^{4}\right),\left(-1,(-1)^{4}\right),\left(-2,(-2)^{4}\right),\left(-3,(-3)^{4}\right)\right\\}$$

Solve each logarithmic equation. Express all solutions in exact form. Support your solutions by using a calculator. $$8 \log \left(4-x^{2}\right)-4=20$$

In \(2012,17 \%\) of the U.S. population was Hispanic, and this number is expected to be \(31 \%\) in \(2060 .\) (Source: U.S. Census Bureau.) (a) Approximate \(C\) and \(a\) so that \(P(x)=C a^{x-2012}\) models these data, where \(P\) is the percent of the population that is Hispanic and \(x\) is the year. Why is \(a>1 ?\) (b) Estimate \(P\) in 2030 . (c) Use \(P\) to estimate the year when \(25 \%\) of the population could be Hispanic.

Explain how the graph of the given function can be obtained from the graph of y=\log _{2} x, and (b) graph the function. $$y=\log _{2}(-x)+1$$

Solve each equation. $$\left(\frac{1}{4}\right)^{2-x}=2^{3 x+3}$$

Use a calculator to find a decimal approximation for each common or natural logarithm. $$\log 28^{3}$$

An everyday activity is described. Keeping in mind that an inverse operation "undoes" what an operation does, describe the inverse activity. tying your shoelaces

Solve each logarithmic equation. Express all solutions in exact form. Support your solutions by using a calculator. $$3 \log _{2}\left(3 x^{2}+2\right)+1=2$$

In \(2012,63 \%\) of the U.S. population was non-Hispanic white, and this number is expected to be \(43 \%\) in \(2060 .\) (Source: U.S. Census Bureau.) (a) Find \(C\) and \(a\) so that \(P(x)=C a^{x-2012}\) models these data, where \(P\) is the percent of the population that is non-Hispanic white and \(x\) is the year. Why is \(a<1 ?\) (b) Estimate \(P\) in 2020 (c) Use \(P\) to estimate when \(50 \%\) of the population could be non-Hispanic white.

Explain how the graph of the given function can be obtained from the graph of y=\log _{2} x, and (b) graph the function. $$y=-\log _{2}(-x)$$

Solve each equation. $$\left(\frac{1}{2}\right)^{3 x-6}=8^{x+1}$$

Use a calculator to find a decimal approximation for each common or natural logarithm. $$\log (47 \times 93)$$

An everyday activity is described. Keeping in mind that an inverse operation "undoes" what an operation does, describe the inverse activity. pressing a car's accelerator

Solve each logarithmic equation. Express all solutions in exact form. Support your solutions by using a calculator. $$2 \log _{2}(5 x-3)+1=17$$

Suppose that the concentration of a bacteria sample is \(100,000\) bacteria per milliliter. If the concentration doubles every 2 hours, how long will it take for the concentration to reach \(350,000\) bacteria per milliliter?

Graph y=\log x^{2} and $y=2 \log x on separate viewing screens. It would seem, at first glance, that by applying the power rule for logarithms, these graphs should be the same. Are they? If not, why not? (Hint: Consider the domain in each case.)

Solve each equation. $$(\sqrt{2})^{x+4}=\left(\frac{1}{4}\right)^{-x}$$

Use a calculator to find a decimal approximation for each common or natural logarithm. $$\ln 43$$