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Using Properties of Logarithms In Exercises \(21-36\) , find the exact value of the logarithmic expression without using a calculator. (If this is not possible, then state the reason.) $$\ln e^{2}+\ln e^{5}$$

The number of bacteria in a culture is increasing according to the law of exponential After 3 hours there are 100 bacteria, and after 5 hours there are 400 bacteria. How many bacteria will there be after 6 hours?

Solve the exponential equation algebraically. Approximate the result to three decimal places. \(6\left(2^{3 x-1}\right)-7=9\)

Using Properties of Logarithms In Exercises \(21-36,\) find the exact value of the logarithmic expression without using a calculator. (If this is not possible, then state the reason.) $$2 \ln e^{6}-\ln e^{5}$$

Graphing an Exponential Function In Exercises \(31-34,\) use a graphing utility to graph the exponential function. $$y=4^{x+1}-2$$

Solve the exponential equation algebraically. Approximate the result to three decimal places. \(8\left(4^{6-2 x}\right)+13=41\)

Sketch the graphs of \(f\) and \(g\) in the same coordinate plane. \(f(x)=5^{x}, g(x)=\log _{5} x\)

Bacteria Growth The number of bacteria in a culture is increasing according to the law of exponential growth. The initial population is 250 bacteria, and the population after 10 hours is double the population after 1 hour. How many bacteria will there be after 6 hours?

Sketch the graphs of \(f\) and \(g\) in the same coordinate plane. \(f(x)=6^{x}, g(x)=\log _{6} x\)

A laptop computer that costs \(\$ 1150\) new has a book value of \(\$ 550\) after 2 years. (a) Find the linear model \(V=m t+b\) (b) Find the exponential model \(V=a e^{k r}\) (c) Use a graphing utility to graph the two models in the same viewing window. Which model depreciates faster in the first 2 years? (d) Find the book values of the computer after 1 year and after 3 years using each model. (e) Explain the advantages and disadvantages of using each model to a buyer and a seller.