91Ó°ÊÓ

The median price of a new home in the United States rose from \(\$ 123,000\) in 1990 to \(\$ 220,000\) in \(2004 .\) Find an exponential function \(P(t)=C e^{t t}\) that models the growth of housing prices, where \(t\) is the number of years since \(1990 .\) (Source: National Association of Home Builders)

State whether each function is one-to-one. $$f(x)=-2 x^{3}+4$$

In Exercises \(31-46,\) write each expression as a logarithm of a single quantity and then simplify if possible. Assume that each variable expression is defined for appropriate values of the variable(s). Do not use a calculator. $$\log 3+\log x+\log \sqrt{y}$$

Evaluate each expression without using a calculator. $$\log _{4} 4^{x^{2}+1}$$

Sketch the graph of each function. $$f(x)=-4(3)^{x}+1$$

Solve the exponential equation. Round to three decimal places, when needed. $$x e^{-x}+e^{x}=2$$

Due to inflation, a dollar in the year 1994 is worth \(\$ 1.28\) in 2005 dollars. Find an exponential function \(v(t)=C e^{k t}\) that models the value of a 1994 dollar t years after \(1994 .\) (Source: Inflationdata.com)

Evaluate each expression without using a calculator. $$\log _{6} 6^{6 x}$$

In Exercises \(31-46,\) write each expression as a logarithm of a single quantity and then simplify if possible. Assume that each variable expression is defined for appropriate values of the variable(s). Do not use a calculator. $$\ln y-\ln 2+\ln \sqrt{x}$$

Solve the exponential equation. Round to three decimal places, when needed. $$e^{x}+e^{-x}=-x+4$$