Chapter 4

Rewrite the equation in terms of base \(e\). Express the answer in terms of a natural logarithm, and then round to three decimal places. $$ y=4.5(0.6)^{x} $$

Solve each logarithmic equation in Exercises \(27-44 .\) Be sure to reject any value of \(x\) that produces the logarithm of a negative number or the logarithm of \(0 .\) $$\log _{2}(4 x+1)=5$$

Evaluate each expression without using a calculator. $$\log _{6} 1$$

Begin by graphing \(f(x)=2^{x} .\) Then use transformations of this graph and a table of coordinates to graph the given function. If applicable, use a graphing utility to confirm your hand-drawn graphs. \(g(x)=\frac{1}{2} \cdot 2^{x}\)

In Exercises \(1-40,\) use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log _{5} \sqrt[3]{\frac{x^{2} y}{25}} $$

Nigeria has a growth rate of 0.031 or \(3.1 \% .\) Describe what this means.

Solve each logarithmic equation in Exercises \(27-44 .\) Be sure to reject any value of \(x\) that produces the logarithm of a negative number or the logarithm of \(0 .\) $$\log _{5} x+\log _{5}(4 x-1)=1$$

Evaluate each expression without using a calculator. $$\log _{5} 5^{7}$$

Graph functions \(f\) and \(g\) in the same rectangular coordinate system. If applicable, use a graphing utility to confirm your hand-drawn graphs. \(f(x)=3^{x}\) and \(g(x)=3^{-x}\)

In Exercises \(1-40,\) use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log _{2} \sqrt[5]{\frac{x y^{4}}{16}} $$

How can you tell if an exponential model describes exponential growth or exponential decay?

Evaluate each expression without using a calculator. $$\log _{4} 4^{6}$$

Graph functions \(f\) and \(g\) in the same rectangular coordinate system. If applicable, use a graphing utility to confirm your hand-drawn graphs. \(f(x)=3^{x}\) and \(g(x)=-3^{x}\)

In Exercises \(1-40,\) use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \ln \left[\frac{x^{3} \sqrt{x^{2}+1}}{(x+1)^{4}}\right] $$

Suppose that a population that is growing exponentially increases from \(800,000\) people in 2003 to \(1,000,000\) people in \(2006 .\) Without showing the details, describeSuppose that a population that is growing exponentially increases from \(800,000\) people in 2003 to \(1,000,000\) people in \(2006 .\) Without showing the details, describe how to obtain the exponential growth function that models the data. how to obtain the exponential growth function that models the data.

Solve each logarithmic equation in Exercises \(27-44 .\) Be sure to reject any value of \(x\) that produces the logarithm of a negative number or the logarithm of \(0 .\) $$\log _{2}(x-1)+\log _{2}(x+1)=3$$

Evaluate each expression without using a calculator. $$8^{\log _{2} 19}$$

Graph functions \(f\) and \(g\) in the same rectangular coordinate system. If applicable, use a graphing utility to confirm your hand-drawn graphs. \(f(x)=3^{x}\) and \(g(x)=\frac{1}{3} \cdot 3^{x}\)

In Exercises \(1-40,\) use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \ln \left[\frac{x^{4} \sqrt{x^{2}+3}}{(x+3)^{5}}\right] $$

Solve each logarithmic equation in Exercises \(27-44 .\) Be sure to reject any value of \(x\) that produces the logarithm of a negative number or the logarithm of \(0 .\) $$\log _{2}(x-1)+\log _{2}(x+1)=3$$

Graph functions \(f\) and \(g\) in the same rectangular coordinate system. If applicable, use a graphing utility to confirm your hand-drawn graphs. \(f(x)=3^{x}\) and \(g(x)=3 \cdot 3^{x}\)

In Exercises \(1-40,\) use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log \left[\frac{10 x^{2} \sqrt[3]{1-x}}{7(x+1)^{2}}\right] $$

Describe a difference between exponential growth and logistic growth.

Solve each logarithmic equation in Exercises \(27-44 .\) Be sure to reject any value of \(x\) that produces the logarithm of a negative number or the logarithm of \(0 .\) $$\log _{2}(x+2)-\log _{2}(x-5)=3$$

Graph \(f(x)=4^{x}\) and \(g(x)=\log _{4} x\) in the same rectangular coordinate system.

Graph functions \(f\) and \(g\) in the same rectangular coordinate system. If applicable, use a graphing utility to confirm your hand-drawn graphs. \(f(x)=\left(\frac{1}{2}\right)^{x}\) and \(g(x)=\left(\frac{1}{2}\right)^{x-1}+1\)

In Exercises \(1-40,\) use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log \left[\frac{100 x^{3} \sqrt[3]{5-x}}{3(x+7)^{2}}\right] $$

Describe the shape of a scatter plot that suggests modeling the data with an exponential function.

Graph \(f(x)=5^{x}\) and \(g(x)=\log _{5} x\) in the same rectangular coordinate system.

Graph functions \(f\) and \(g\) in the same rectangular coordinate system. If applicable, use a graphing utility to confirm your hand-drawn graphs. \(f(x)=\left(\frac{1}{2}\right)^{x}\) and \(g(x)=\left(\frac{1}{2}\right)^{x-1}+2\)

In Exercises \(41-70\), use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(I .\) Where possible, evaluate logarithmic expressions. $$ \log 5+\log 2 $$

Solve each logarithmic equation in Exercises \(27-44 .\) Be sure to reject any value of \(x\) that produces the logarithm of a negative number or the logarithm of \(0 .\) $$2 \log _{3}(x+4)=\log _{3} 9+2$$

Graph \(f(x)=\left(\frac{1}{2}\right)^{x}\) and \(g(x)=\log _{1 / 2} x\) in the same rectangular coordinate system.

Use the compound interest formulas \(A=P\left(1+\frac{r}{n}\right)^{n t}\) and \(A=P e^{r t}\) to solve. Round answers to the nearest cent. Find the accumulated value of an investment of \(\$ 10,000\) for 5 years at an interest rate of \(5.5 \%\) if the money is a. compounded semiannually; b. compounded quarterly; c. compounded monthly; d. compounded continuously.

In Exercises \(41-70\), use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(I .\) Where possible, evaluate logarithmic expressions. $$ \log 250+\log 4 $$

You take up weightlifting and record the maximum number of pounds you can lift at the end of each week. You start off with rapid growth in terms of the weight you can lift from week to week, but then the growth begins to level off. Describe how to obtain a function that models the number of pounds you can lift at the end of each week. How can you use this function to predict what might happen if you continue the sport?

Solve each logarithmic equation in Exercises \(27-44 .\) Be sure to reject any value of \(x\) that produces the logarithm of a negative number or the logarithm of \(0 .\) $$3 \log _{2}(x-1)=5-\log _{2} 4$$

Graph \(f(x)=\left(\frac{1}{4}\right)^{x}\) and \(g(x)=\log _{1 / 4} x\) in the same rectangular coordinate system.

Use the compound interest formulas \(A=P\left(1+\frac{r}{n}\right)^{n t}\) and \(A=P e^{r t}\) to solve. Round answers to the nearest cent. Find the accumulated value of an investment of \(\$ 5000\) for 10 years at an interest rate of \(6.5 \%\) if the money is a. compounded semiannually; b. compounded quarterly; c. compounded monthly; d. compounded continuously.

In Exercises \(41-70\), use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(I .\) Where possible, evaluate logarithmic expressions. $$ \ln x+\ln 7 $$

Would you prefer that your salary be modeled exponentially or logarithmically? Explain your answer.

Solve each logarithmic equation in Exercises \(27-44 .\) Be sure to reject any value of \(x\) that produces the logarithm of a negative number or the logarithm of \(0 .\) $$\log _{2}(x-6)+\log _{2}(x-4)-\log _{2} x=2$$

Use the compound interest formulas \(A=P\left(1+\frac{r}{n}\right)^{n t}\) and \(A=P e^{r t}\) to solve. Round answers to the nearest cent. Suppose that you have \(\$ 12,000\) to invest. Which investment yields the greatest return over 3 years: \(7 \%\) compounded monthly or \(6.85 \%\) compounded continuously?

In Exercises \(41-70\), use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(I .\) Where possible, evaluate logarithmic expressions. $$ \ln x+\ln 3 $$

One problem with all exponential growth models is that nothing can grow exponentially forever. Describe factors that might limit the size of a population.

Solve each logarithmic equation in Exercises \(27-44 .\) Be sure to reject any value of \(x\) that produces the logarithm of a negative number or the logarithm of \(0 .\) $$\log _{2}(x-3)+\log _{2} x-\log _{2}(x+2)=2$$

Use the compound interest formulas \(A=P\left(1+\frac{r}{n}\right)^{n t}\) and \(A=P e^{r t}\) to solve. Round answers to the nearest cent. Suppose that you have \(\$ 6000\) to invest. Which investment yields the greatest return over 4 years: \(8.25 \%\) compounded quarterly or \(8.3 \%\) compounded semiannually?

In Exercises \(41-70\), use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(I .\) Where possible, evaluate logarithmic expressions. $$ \log _{2} 96-\log _{2} 3 $$

Exercises \(45-52\) involve equations with natural logarithms. Solve each equation by isolating the natural logarithm and exponentiating both sides. Express the answer in terms of \(e\) Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$\ln x=2$$

The exponential function \(f(x)=67.38(1.026)^{x}\) describes the population of Mexico, \(f(x),\) in millions, \(x\) years after 1980 a. Substitute 0 for \(x\) and, without using a calculator, find Mexico's population in 1980 . b. Substitute 27 for \(x\) and use your calculator to find Mexico's population in the year 2007 as predicted by this function. c. Find Mexico's population in the year 2034 as predicted by this function. d. Find Mexico's population in the year 2061 as predicted by this function. e. What appears to be happening to Mexico's population every 27 years?