Chapter 4

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. $$ \log \sqrt[5]{\frac{x}{y}} $$

In Exercises \(25-34,\) begin by graphing \(f(x)-2^{x}\). Then use transformations of this graph to graph the given function. Be sure to graph and give equations of the asymptotes Use the graphs to determine each function's domain and range. If applicable, use a graphing utility to confirm your hand-drawn graphs. $$ g(x)-2^{-x} $$

Use the exponential growth model, \(A=A_{0} e^{t_{i}},\) to show that the time it takes a population to double (to grow from \(A_{0}\) to \(2 A_{0}\) ) is given by \(t=\frac{\ln 2}{k}\)..

Evaluate each expression without using a calculator. $$\log _{64} 8$$

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. $$ \log _{b}\left(\frac{\sqrt{x} y^{3}}{z^{3}}\right) $$

In Exercises \(25-34,\) begin by graphing \(f(x)-2^{x}\). Then use transformations of this graph to graph the given function. Be sure to graph and give equations of the asymptotes Use the graphs to determine each function's domain and range. If applicable, use a graphing utility to confirm your hand-drawn graphs. $$ g(x)-2 \cdot 2^{x} $$

Solve each exponential equation in Exercises \(23-48\). Express the solution set in terms of natural logarithms or common logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$ e^{1-8 x}-7957 $$

Evaluate each expression without using a calculator. $$\log _{81} 9$$

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. $$ \log _{b}\left(\frac{\sqrt[3]{x} y^{4}}{z^{5}}\right) $$

Solve each exponential equation in Exercises \(23-48\). Express the solution set in terms of natural logarithms or common logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$ e^{5 x-3}-2-10.476 $$

Use the formula \(t-\frac{\ln 2}{k}\) that gives the time for a population with a growth rate \(k\) to double to solve Exercises \(35-36 .\) Express each answer to the nearest whole year. The growth model \(A-4.3 e^{0.01 t}\) describes New Zealand's population, \(A,\) in millions, \(t\) years after 2010 . a. What is New Zealand's growth rate? b. How long will it take New Zealand to double its population?

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

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. $$ \log _{5} \sqrt[7]{\frac{x^{2} y}{25}} $$

Use the formula \(t-\frac{\ln 2}{k}\) that gives the time for a population with a growth rate \(k\) to double to solve Exercises \(35-36 .\) Express each answer to the nearest whole year. The growth model \(A=112.5 e^{0.012}\) describes Mexico's population, \(A,\) in millions, \(t\) years after 2010 . a. What is Mexico's growth rate? b. How long will it take Mexico to double its population?

Evaluate each expression without using a calculator. $$\log _{11} 11$$

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. $$ \log _{2} \sqrt[5]{\frac{x y^{4}}{16}} $$

Solve each exponential equation in Exercises \(23-48\). Express the solution set in terms of natural logarithms or common logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$ 7^{x+2}-410 $$

The logistic growth function \(f(t)=\frac{100,000}{1+5000 e^{-t}}\) describes the number of people, \(f(t)\), who have become ill with influenza \(t\) weeks after its initial outbreak in a particular community. a. How many people became ill with the flu when the epidemic began? b. How many people were ill by the end of the fourth week? c. What is the limiting size of the population that becomes ill?

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

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. $$ \ln \left[\frac{x^{3} \sqrt{x^{2}+1}}{(x+1)^{4}}\right] $$

Solve each exponential equation in Exercises \(23-48\). Express the solution set in terms of natural logarithms or common logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$ 5^{x-3}-137 $$

Shown, again, in the following table is world population, in billions, for seven selected years from 1950 through \(2010 .\) Using a graphing utility's logistic regression option, we obtain the equation shown on the screen. (TABLE CANNOT COPY) We see from the calculator screen at the bottom of the previous page that a logistic growth model for world population, \(f(x),\) in billions, \(x\) years after 1949 is $$f(x)=\frac{12.57}{1+4.11 e^{-0.52 h x}}$$ Use this function to solve Exercises \(38-42\) How well does the function model the data showing a world population of 6.1 billion for \(2000 ?\)

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

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. $$ \ln \left[\frac{x^{4} \sqrt{x^{2}+3}}{(x+3)^{5}}\right] $$

Solve each exponential equation in Exercises \(23-48\). Express the solution set in terms of natural logarithms or common logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$ 7^{0.3 x}-813 $$

Shown, again, in the following table is world population, in billions, for seven selected years from 1950 through \(2010 .\) Using a graphing utility's logistic regression option, we obtain the equation shown on the screen. (TABLE CANNOT COPY) We see from the calculator screen at the bottom of the previous page that a logistic growth model for world population, \(f(x),\) in billions, \(x\) years after 1949 is $$f(x)=\frac{12.57}{1+4.11 e^{-0.52 h x}}$$ Use this function to solve Exercises \(38-42\) How well does the function model the data showing a world population of 6.9 billion for \(2010 ?\)

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

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. $$ \log \left[\frac{10 x^{2} \sqrt[3]{1-x}}{7(x+1)^{2}}\right] $$

Shown, again, in the following table is world population, in billions, for seven selected years from 1950 through \(2010 .\) Using a graphing utility's logistic regression option, we obtain the equation shown on the screen. (TABLE CANNOT COPY) We see from the calculator screen at the bottom of the previous page that a logistic growth model for world population, \(f(x),\) in billions, \(x\) years after 1949 is $$f(x)=\frac{12.57}{1+4.11 e^{-0.52 h x}}$$ Use this function to solve Exercises \(38-42\) When will world population reach 7 billion?

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

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. $$ \log \left[\begin{array}{c} 100 x^{3} \sqrt[3]{5-x} \\ 3(x+7)^{2} \end{array}\right] $$

Shown, again, in the following table is world population, in billions, for seven selected years from 1950 through \(2010 .\) Using a graphing utility's logistic regression option, we obtain the equation shown on the screen. (TABLE CANNOT COPY) We see from the calculator screen at the bottom of the previous page that a logistic growth model for world population, \(f(x),\) in billions, \(x\) years after 1949 is $$f(x)=\frac{12.57}{1+4.11 e^{-0.52 h x}}$$ Use this function to solve Exercises \(38-42\) When will world population reach 8 billion?

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

In Exercises \(41-70\), use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(1 .\) Where possible, evaluate logarithmic expressions without using a calculator. $$ \log 5+\log 2 $$

Solve each exponential equation in Exercises \(23-48\). Express the solution set in terms of natural logarithms or common logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$ 7^{2 x+1}-3^{x+2} $$

Shown, again, in the following table is world population, in billions, for seven selected years from 1950 through \(2010 .\) Using a graphing utility's logistic regression option, we obtain the equation shown on the screen. (TABLE CANNOT COPY) We see from the calculator screen at the bottom of the previous page that a logistic growth model for world population, \(f(x),\) in billions, \(x\) years after 1949 is $$f(x)=\frac{12.57}{1+4.11 e^{-0.52 h x}}$$ Use this function to solve Exercises \(38-42\) According to the model, what is the limiting size of the population that Earth will eventually sustain?

Evaluate each expression without using a calculator. $$7^{\log _{7} 23}$$

In Exercises \(41-70\), use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(1 .\) Where possible, evaluate logarithmic expressions without using a calculator. $$ \log 250+\log 4 $$

The logistic growth function $$P(x)=\frac{90}{1+271 e^{-0.122 x}}$$ models the percentage, \(P(x),\) of Americans who are \(x\) years old with some coronary heart disease. Use the function to solve Exercises \(43-46\) What percentage of 20 -year-olds have some coronary heart disease?

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

In Exercises \(41-70,\) use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(1 .\) Where possible, evaluate logarithmic expressions without using a calculator. $$ \ln x+\ln 7 $$

Solve each exponential equation in Exercises \(23-48\). Express the solution set in terms of natural logarithms or common logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$ e^{2 x}-2 e^{x}-3-0 $$

The logistic growth function $$P(x)=\frac{90}{1+271 e^{-0.122 x}}$$ models the percentage, \(P(x),\) of Americans who are \(x\) years old with some coronary heart disease. Use the function to solve Exercises \(43-46\) What percentage of 80 -year-olds have some coronary heart disease?

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

In Exercises \(41-70,\) use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(1 .\) Where possible, evaluate logarithmic expressions without using a calculator. $$ \ln x+\ln 3 $$

The logistic growth function $$P(x)=\frac{90}{1+271 e^{-0.122 x}}$$ models the percentage, \(P(x),\) of Americans who are \(x\) years old with some coronary heart disease. Use the function to solve Exercises \(43-46\) At what age is the percentage of some coronary heart disease \(50 \% ?\)

In Exercises \(41-70,\) use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(1 .\) Where possible, evaluate logarithmic expressions without using a calculator. $$ \log _{2} 96-\log _{2} 3 $$

The logistic growth function $$P(x)=\frac{90}{1+271 e^{-0.122 x}}$$ models the percentage, \(P(x),\) of Americans who are \(x\) years old with some coronary heart disease. Use the function to solve Exercises \(43-46\) At what age is the percentage of some coronary heart disease \(70 \% ?\)

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

In Exercises \(41-70,\) use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(1 .\) Where possible, evaluate logarithmic expressions without using a calculator. $$ \log _{3} 405-\log _{3} 5 $$