Chapter 4

In Exercises \(21-30,\) write each logarithm as a sum and\or difference of logarithmic expressions. Eliminate exponents and radicals and evaluate logarithms wherever possible. Assume that \(a, x, y\) \(z>0\) and \(a \neq 1\). $$\log \sqrt[3]{\frac{x y^{3}}{z^{5}}}$$

Evaluate each expression without using a calculator. $$\log _{3} \frac{1}{81}$$

Sketch the graph of each function. $$f(x)=-2\left(\frac{1}{3}\right)^{x}$$

Solve the exponential equation. Round to three decimal places, when needed. $$10^{2 x^{2}+1}-8=4$$

The population of the United States is expected to grow from 282 million in 2000 to 335 million in \(2020 .\) (Source: U.S. Census Bureau) (a) Find a function of the form \(P(t)=C e^{k t}\) that models the population growth of the United States. Here, \(t\) is the number of years since 2000 and \(P(t)\) is in millions. (b) Assuming the trend in part (a) continues, in what year will the population of the United States be 300 million?

state whether each function is one-to-one. $$f(x)=\frac{4}{3} x+1$$

In Exercises \(21-30,\) write each logarithm as a sum and\or difference of logarithmic expressions. Eliminate exponents and radicals and evaluate logarithms wherever possible. Assume that \(a, x, y\) \(z>0\) and \(a \neq 1\). $$\log \sqrt[3]{\frac{x^{3} z^{5}}{10 y^{2}}}$$

Evaluate each expression without using a calculator. $$\log _{7} \frac{1}{49}$$

Sketch the graph of each function. $$h(x)=4\left(\frac{2}{3}\right)^{x}$$

Solve the exponential equation. Round to three decimal places, when needed. $$1.7 e^{0.5 x}=3.26$$

The population of Florida grew from 16.0 million in 2000 to 17.4 million in \(2004 .\) (Source: U.S. Census Bureau) (a) Find a function of the form \(P(t)=C e^{t x}\) that models the population growth. Here, \(t\) is the number of years since 2000 and \(P(t)\) is in millions. (b) Use your model to predict the population of Florida in 2010.

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

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. $$\log 6.3-\log 3$$

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

Sketch the graph of each function. $$f(x)=3^{2 x}$$

Solve the exponential equation. Round to three decimal places, when needed. $$4 e^{x}=-x+3$$

State whether each function is one-to-one. $$f(x)=-3 x^{2}+1$$

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. $$\log 4.1+\log 3$$

Evaluate each expression without using a calculator. $$\log _{1 / 3} 9$$

Sketch the graph of each function. $$g(x)=2^{3 x}$$

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

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. $$e^{x}+e^{-x}=-x+4$$

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)

State whether each function is one-to-one. $$f(x)=-\frac{1}{3} x^{3}-5$$

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. $$\ln y-\ln 2+\ln \sqrt{x}$$

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

Sketch the graph of each function. $$f(x)=-2(3)^{x}+1$$

Solve the logarithmic equation and eliminate any extraneous solutions. If there are no solutions, so state. $$\log x=0$$

The purchase price of a 2006 Ford \(\mathrm{F} 150\) longbed pickup truck is \(\$ 23,024 .\) After 1 year, the price of the Ford \(\mathrm{F} 150\) is \(\$ 17,160 .\) (Source: Kelley Blue Book) (a) Find an exponential function, \(P(t)=C e^{k t},\) that models the price of the truck, where \(t\) is the number of years since 2006 (b) What will be the value of the Ford \(\mathrm{F} 150\) in the year \(2009 ?\)

Find the inverse of the given function. Then graph the given function and its inverse on the same set of axes. $$f(x)=-\frac{2}{3} x$$

Evaluate the expression to four decimal places using a calculator. $$2 \log 4$$

Sketch the graph of each function. $$f(x)=2^{-x}-1$$

Solve the logarithmic equation and eliminate any extraneous solutions. If there are no solutions, so state. $$\ln x=1$$

The spread of the flu in an elementary school can be modeled by a logistic function. The number of children infected with the flu virus \(t\) days after the first infection is given by $$N(t)=\frac{150}{1+4 e^{-0.5 t^{2}}}$$ (a) How many children were initially infected with the flu? (b) How many children were infected with the flu virus after 5 days? after 10 days?

Find the inverse of the given function. Then graph the given function and its inverse on the same set of axes. $$g(x)=\frac{4}{3} x$$

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. $$\ln 4-1$$

Evaluate the expression to four decimal places using a calculator. $$-3 \log 6$$

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

Evaluate the expression to four decimal places using a calculator. $$\ln \sqrt{2}$$

Find the inverse of the given function. Then graph the given function and its inverse on the same set of axes. $$f(x)=-4 x+\frac{1}{5}$$

Solve the logarithmic equation and eliminate any extraneous solutions. If there are no solutions, so state. $$\ln (x-1)=2$$

The population of white-tailed deer in a wildlife refuge \(t\) months after their introduction into the refuge can be modeled by the logistic function $$N(t)=\frac{300}{1+14 e^{-0.05 t^{2}}}$$ (a) How many deer were initially introduced into the refuge? (b) How many deer will be in the wildlife refuge 10 months after introduction?

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. $$\log 8+1$$

Sketch the graph of each function and find (a) the \(y\) -intercept; (b) the domain and range; (c) the horizontal asymptote;and (d) the behavior of the function as \(x\) approaches\(\pm \infty .\) $$f(x)=-5^{x}$$

Evaluate the expression to four decimal places using a calculator. $$\ln \pi$$