91Ó°ÊÓ

The hydrogen ion concentration of a sample of each substance is given. Calculate the \(\mathrm{pH}\) of the substance. (a) Lemon juice: \(\left[\mathrm{H}^{+}\right]=5.0 \times 10^{-3} \mathrm{M}\) (b) Tomato juice: \(\left[\mathrm{H}^{+}\right]=3.2 \times 10^{-4} \mathrm{M}\) (c) Seawater: \(\left[\mathrm{H}^{+}\right]=5.0 \times 10^{-9} \mathrm{M}\)

A certain species of bird was introduced in a certain county 25 years ago. Biologists observe that the population doubles every 10 years, and now the population is \(13,000\) (a) What was the initial size of the bird population? (b) Estimate the bird population 5 years from now. (c) Sketch a graph of the bird population.

The count in a culture of bacteria was 400 after 2 hours and \(25,600\) after 6 hours. (a) What is the relative rate of growth of the bacteria population? Express your answer as a percentage. (b) What was the initial size of the culture? (c) Find a function that models the number of bacteria \(n(t)\) after \(t\) hours. (d) Find the number of bacteria after 4.5 hours. (e) After how many hours will the number of bacteria reach \(50.000 ?\)

Express the equation in logarithmic form. (a) \(10^{4}=10,000\) (b) \(5^{-2}=\frac{1}{25}\)

Find the exact solution of the exponential equation in terms of logarithms. (b) Use a calculator to find an approximation to the solution rounded to six decimal places. $$2^{3 x+1}=3^{x-2}$$

Find the exact solution of the exponential equation in terms of logarithms. (b) Use a calculator to find an approximation to the solution rounded to six decimal places. $$7^{x / 2}=5^{1-x}$$

Compare the graphs of the power function \(f\) and exponential function \(g\) by evaluating both of them for \(x=0,1,2,3,4,6,8,\) and 10 Then draw the graphs of \(f\) and \(g\) on the same set of axes. \(f(x)=x^{4} ; \quad g(x)=4^{x}\)