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. $$ \log _{5}(7 \cdot 3) $$

The exponential growth model \(A=203 e^{0.011}\) describes the population of the United States, \(A,\) in millions, \(t\) years after \(1970 . What was the population of the United States in \)1970 ?$

Solve each exponential equation in Exercises \(1-26\) Express the solution set in terms of natural logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$10^{x}=3.91$$

Write each equation in its equivalent exponential form. $$4=\log _{2} 16$$

Approximate each number using a calculator. Round your answer to three decimal places. \(2^{3.4}\)

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. $$ \log _{8}(13 \cdot 7) $$

Solve each exponential equation in Exercises \(1-26\) Express the solution set in terms of natural logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$10^{x}=8.07$$

Write each equation in its equivalent exponential form. $$6=\log _{2} 64$$

Approximate each number using a calculator. Round your answer to three decimal places. \(3^{2.4}\)

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. $$ \log _{7}(7 x) $$

Solve each exponential equation in Exercises \(1-26\) Express the solution set in terms of natural logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$e^{x}=5.7$$

Write each equation in its equivalent exponential form. $$2=\log _{3} x$$

Approximate each number using a calculator. Round your answer to three decimal places. \(3^{\sqrt{5}}\)

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. $$ \log _{9}(9 x) $$

Solve each exponential equation in Exercises \(1-26\) Express the solution set in terms of natural logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$e^{x}=0.83$$

Write each equation in its equivalent exponential form. $$2=\log _{9} x$$

Approximate each number using a calculator. Round your answer to three decimal places. \(5^{\sqrt{3}}\)

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. $$ \log (1000 x) $$

India is currently one of the world's fastest-growing countries. By \(2040,\) the population of India will be larger than the population of China; by 2050 , nearly one-third of the world's population will live in these two countries alone. The exponential growth model \(A=574 e^{0.036 t}\) describes the population of India, \(A,\) in millions, \(t\) years after \(1974 .\) By what percentage is the population of India increasing each year?

Solve each exponential equation in Exercises \(1-26\) Express the solution set in terms of natural logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$5^{x}=17$$

Write each equation in its equivalent exponential form. $$5=\log _{b} 32$$

Approximate each number using a calculator. Round your answer to three decimal places. \(4^{-1.5}\)

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. $$ \log (10,000 x) $$

India is currently one of the world's fastest-growing countries. By \(2040,\) the population of India will be larger than the population of China; by 2050 , nearly one-third of the world's population will live in these two countries alone. The exponential growth model \(A=574 e^{0.036 t}\) describes the population of India, \(A,\) in millions, \(t\) years after \(1974 .\) What was the population of India in \(1974 ?\)

Solve each exponential equation in Exercises \(1-26\) Express the solution set in terms of natural logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$19^{x}=143$$

Write each equation in its equivalent exponential form. $$3=\log _{b} 27$$

Approximate each number using a calculator. Round your answer to three decimal places. \(6^{-1.2}\)

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. $$ \log _{7}\left(\frac{7}{x}\right) $$

India is currently one of the world's fastest-growing countries. By \(2040,\) the population of India will be larger than the population of China; by 2050 , nearly one-third of the world's population will live in these two countries alone. The exponential growth model \(A=574 e^{0.036 t}\) describes the population of India, \(A,\) in millions, \(t\) years after \(1974 .\) When will India's population be 1624 million?

Solve each exponential equation in Exercises \(1-26\) Express the solution set in terms of natural logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$5 e^{x}=23$$

Write each equation in its equivalent exponential form. $$\log _{6} 216=y$$

Approximate each number using a calculator. Round your answer to three decimal places. \(e^{2.3}\)

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. $$ \log _{9}\left(\frac{9}{x}\right) $$

India is currently one of the world's fastest-growing countries. By \(2040,\) the population of India will be larger than the population of China; by 2050 , nearly one-third of the world's population will live in these two countries alone. The exponential growth model \(A=574 e^{0.036 t}\) describes the population of India, \(A,\) in millions, \(t\) years after \(1974 .\) When will India's population be 2732 million?

Solve each exponential equation in Exercises \(1-26\) Express the solution set in terms of natural logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$9 e^{x}=107$$

Write each equation in its equivalent exponential form. $$\log _{5} 125=y$$

Approximate each number using a calculator. Round your answer to three decimal places. \(e^{3.4}\)

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. $$ \log \left(\frac{x}{100}\right) $$

Solve each exponential equation in Exercises \(1-26\) Express the solution set in terms of natural logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$3 e^{5 x}=1977$$

Write each equation in its equivalent logarithmic form. $$2^{3}=8$$

Approximate each number using a calculator. Round your answer to three decimal places. \(e^{-0.95}\)

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. $$ \log \left(\frac{x}{1000}\right) $$

India is currently one of the world's fastest-growing countries. By \(2040,\) the population of India will be larger than the population of China; by 2050 , nearly one-third of the world's population will live in these two countries alone. The exponential growth model \(A=574 e^{0.036 t}\) describes the population of India, \(A,\) in millions, \(t\) years after \(1974 .\) In \(2000,\) the population of the Palestinians in the West Bank, Gaza Strip, and East Jerusalem was approximately 3.2 million and by 2050 it is projected to grow to 12 million. Use the exponential growth model \(A=A_{0} e^{k t},\) in which \(t\) is the number of years after \(2000,\) to find the exponential growth function that models the data.

Solve each exponential equation in Exercises \(1-26\) Express the solution set in terms of natural logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$4 e^{7 x}=10,273$$

Write each equation in its equivalent logarithmic form. $$5^{4}=625$$

Approximate each number using a calculator. Round your answer to three decimal places. \(e^{-0.75}\)

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. $$ \log _{4}\left(\frac{64}{y}\right) $$

India is currently one of the world's fastest-growing countries. By \(2040,\) the population of India will be larger than the population of China; by 2050 , nearly one-third of the world's population will live in these two countries alone. The exponential growth model \(A=574 e^{0.036 t}\) describes the population of India, \(A,\) in millions, \(t\) years after \(1974 .\) In \(2000,\) the population of Israel was approximately 6.04 million and by 2050 it is projected to grow to 10 million. Use the exponential growth model \(A=A_{0} e^{k t},\) in which \(t\) is the number of years after \(2000,\) to find an exponential growth function that models the data.

Solve each exponential equation in Exercises \(1-26\) Express the solution set in terms of natural logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$e^{1-5 x}=793$$

Write each equation in its equivalent logarithmic form. $$2^{-4}=\frac{1}{16}$$