Chapter 4

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 models describe the population of the indicated country, \(A,\) in millions, t years after \(2010 .\) Use these models to solve Exercises \(1-6\) $$ \begin{aligned} &India \quad A=1173.1 e^{0.008 t}\\\ &Iraq \quad A=31.5 e^{0.019 t}\\\ &Japan \quad A=127.3 e^{-0.006 t}\\\ &Russia \quad A=141.9 e^{-0.005 t} \end{aligned} $$ What was the population of Japan in \(2010 ?\)

Solve each exponential equation by expressing each side as a power of the same base and then equating exponents. $$ 2^{x}=64 $$

In Exercises 1–8, 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. $$ 2^{3.4} $$

Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log _{8}(13 \cdot 7) $$

The exponential models describe the population of the indicated country, \(A,\) in millions, t years after \(2010 .\) Use these models to solve Exercises \(1-6\) $$ \begin{aligned} &India \quad A=1173.1 e^{0.008 t}\\\ &Iraq \quad A=31.5 e^{0.019 t}\\\ &Japan \quad A=127.3 e^{-0.006 t}\\\ &Russia \quad A=141.9 e^{-0.005 t} \end{aligned} $$ What was the population of Iraq in \(2010 ?\)

Solve each exponential equation by expressing each side as a power of the same base and then equating exponents. $$ 3^{x}=81 $$

In Exercises 1–8, 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. $$ 3^{2.4} $$

Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log _{7}(7 x) $$

The exponential models describe the population of the indicated country, \(A,\) in millions, t years after \(2010 .\) Use these models to solve Exercises \(1-6\) $$ \begin{aligned} &India \quad A=1173.1 e^{0.008 t}\\\ &Iraq \quad A=31.5 e^{0.019 t}\\\ &Japan \quad A=127.3 e^{-0.006 t}\\\ &Russia \quad A=141.9 e^{-0.005 t} \end{aligned} $$ Which country has the greatest growth rate? By what percentage is the population of that country increasing each year?

Solve each exponential equation by expressing each side as a power of the same base and then equating exponents. $$ 5^{x}=125 $$

In Exercises 1–8, 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. $$ 3^{\sqrt{5}} $$

Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log _{9}(9 x) $$

The exponential models describe the population of the indicated country, \(A,\) in millions, t years after \(2010 .\) Use these models to solve Exercises \(1-6\) $$ \begin{aligned} &India \quad A=1173.1 e^{0.008 t}\\\ &Iraq \quad A=31.5 e^{0.019 t}\\\ &Japan \quad A=127.3 e^{-0.006 t}\\\ &Russia \quad A=141.9 e^{-0.005 t} \end{aligned} $$ Which countries have a decreasing population? By what percentage is the population of these countries decreasing each year?

Solve each exponential equation by expressing each side as a power of the same base and then equating exponents. $$ 5^{x}=625 $$

In Exercises 1–8, 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. $$ 5^{\sqrt{3}} $$

Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log (1000 x) $$

The exponential models describe the population of the indicated country, \(A,\) in millions, t years after \(2010 .\) Use these models to solve Exercises \(1-6\) $$ \begin{aligned} &India \quad A=1173.1 e^{0.008 t}\\\ &Iraq \quad A=31.5 e^{0.019 t}\\\ &Japan \quad A=127.3 e^{-0.006 t}\\\ &Russia \quad A=141.9 e^{-0.005 t} \end{aligned} $$ When will India's population be 1377 million?

Solve each exponential equation by expressing each side as a power of the same base and then equating exponents. $$ 2^{2 x-1}=32 $$

In Exercises 1–8, 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. $$ 4^{-1.5} $$

Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log (10,000 x) $$

The exponential models describe the population of the indicated country, \(A,\) in millions, t years after \(2010 .\) Use these models to solve Exercises \(1-6\) $$ \begin{aligned} &India \quad A=1173.1 e^{0.008 t}\\\ &Iraq \quad A=31.5 e^{0.019 t}\\\ &Japan \quad A=127.3 e^{-0.006 t}\\\ &Russia \quad A=141.9 e^{-0.005 t} \end{aligned} $$ When will India's population be 1491 million?

Solve each exponential equation by expressing each side as a power of the same base and then equating exponents. $$ 3^{2 x+1}=27 $$

In Exercises 1–8, 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. $$ 6^{-1.2} $$

Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator.$$\log _{7}\left(\frac{7}{x}\right)$$.

About the size of New Jersey, Israel has seen its population soar to more than 6 million since it was established. The graphs show that by \(2050,\) Palestinians in the West Bank, Gaza Strip, and East Jerusalem will outnumber Israelis. Exercises \(7-8\) involve the projected growth of these two populations. (Graph can't copy) a. 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. b. In which year will Israel's population be 9 million?

Solve each exponential equation by expressing each side as a power of the same base and then equating exponents. $$ 4^{2 x-1}=64 $$

In Exercises 1–8, 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. $$ e^{2,3} $$

About the size of New Jersey, Israel has seen its population soar to more than 6 million since it was established. The graphs show that by \(2050,\) Palestinians in the West Bank, Gaza Strip, and East Jerusalem will outnumber Israelis. Exercises \(7-8\) involve the projected growth of these two populations. (Graph can't copy) a. 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. b. In which year will the Palestinian population be 9 million?

Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log _{9}\left(\frac{9}{x}\right) $$

Solve each exponential equation by expressing each side as a power of the same base and then equating exponents. $$ 5^{3 x-1}=125 $$

In Exercises 1–8, 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. $$ e^{3.4} $$

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 by expressing each side as a power of the same base and then equating exponents. $$ 32^{x}=8 $$

In Exercises 9–20, write each equation in its equivalent logarithmic form. $$ 2^{3}=8 $$

approximate each number using a calculator. Round your answer to three decimal places. $$ e^{-0.95} $$

Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log \left(\frac{x}{1000}\right) $$

Solve each exponential equation by expressing each side as a power of the same base and then equating exponents. $$ 4^{x}=32 $$

In Exercises 9–20, write each equation in its equivalent logarithmic form. $$ 5^{4}=625 $$

approximate each number using a calculator. Round your answer to three decimal places. $$ e^{-0.75} $$

Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log _{4}\left(\frac{64}{y}\right) $$

Solve each exponential equation by expressing each side as a power of the same base and then equating exponents. $$ 9^{x}=27 $$