91Ó°ÊÓ

Express in terms of sums and differences of logarithms. $$\log _{b} \frac{x^{2} y}{b^{3}}$$

The Beer-Lambert Law. A beam of light enters a medium such as water or smog with initial intensity \(I_{0} .\) Its intensity decreases depending on the thickness (or concentration) of the medium. The intensity \(I\) at a depth (or concentration) of \(x\) units is given by $$I=I_{0} e^{-\mu x}$$, The constant \(\mu\) (the Greek letter "mu") is called the coefficient of absorption, and it varies with the medium. For sea water, \(\mu=1.4\) a) What percentage of light intensity \(I_{0}\) remains in sea water at a depth of \(1 \mathrm{m} ? 3 \mathrm{m} ? 5 \mathrm{m} ?\) \(50 \mathrm{m} ?\) b) Plant life cannot exist below \(10 \mathrm{m}\). What percentage of \(I_{0}\) remains at \(10 \mathrm{m} ?\)

Sketch the graph of the function and check the graph with a graphing calculator. Describe how each graph can be obtained from the graph of a basic exponential function. $$y=e^{2 x}+1$$

Compound Interest. Suppose that \(\$ 82.000\) is invested at \(4 \frac{1}{2} \%\) interest, compounded quarterly. a) Find the function for the amount to which the investment grows after \(t\) years. b) Graph the function. c) Find the amount of money in the account at \(t=0,2\) \(5,\) and 10 years. d) When will the amount of money in the account reach \(\$ 100,000 ?\)

Graph the function and its inverse using a graphing calculator. Use an inverse drawing feature, if available. Find the domain and the range of \(f\) and of \(f^{-1}\). $$f(x)=3-x^{2}, x \geq 0$$

Compound Interest. Suppose that \(\$ 750\) is invested at \(7 \%\) interest, compounded semiannually. a) Find the function for the amount to which the investment grows after \(t\) years. b) Graph the function. c) Find the amount of money in the account at \(t=1,6\) \(10,15,\) and 25 years. d) When will the amount of money in the account reach \(\$ 3000 ?\)

Given that \(\log _{a} 2=0.301, \log _{a} 7=0.845,\) and \(\log _{a} 11=1.041,\) find each of the following, if possible. Round the answer to the nearest thousandth. $$\log _{a} 98$$

Given that \(\log _{b} 2=0.693, \log _{b} 3=1.099,\) and \(\log _{b} 5=1.609,\) find each of the following, if possible. Round the answer to the nearest thousandth. $$\log _{b} 125$$

Find the following using a calculator. Round to four decimal places. $$\ln 50$$

Centenarian Population. The centenarian population in the United States has grown over \(65 \%\) in the last 30 years. In \(1980,\) there were only \(32,194\) residents ages 100 and over. This number had grown to \(53,364\) by \(2010 .\) (Sources: Population Projections Program; U.S. Census Bureau; U.S. Department of Commerce; "What People Who Live to 100 Have in Common," by Emily Brandon, U.S. News and World Report, January \(7,2013\) ) The exponential function $$ H(t)=80,040.68(1.0481)^{t} $$ where \(t\) is the number of years after \(2015,\) can be used to project the number of centenarians. Use this function to project the centenarian population in 2020 and in 2050 (IMAGE CANT COPY)