Chapter 4

Use properties of logarithms to expand logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. \(\ln \left(\frac{e^{4}}{8}\right)\)

An artifact originally had 16 grams of carbon- 14 present. The decay model \(A=16 e^{-0.0001211}\) describes the amount of carbon-I4 present after t years. Use this model to solve Exercises \(15-16 .\) How many grams of carbon-14 will be present in 5715 years?

Solve each exponential equation by expressing each side as a power of the same base and then equating exponents. $$6^{\frac{x-3}{4}}=\sqrt{6}$$

Write each equation in its equivalent logarithmic form. $$13^{2}=x$$

Graph each function by making a table of coordinates. If applicable, use a graphing utility to confirm your hand-drawn graph. $$h(x)=\left(\frac{1}{2}\right)^{x}$$

Use properties of logarithms to expand logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. \(\log _{b} x^{3}\)

An artifact originally had 16 grams of carbon- 14 present. The decay model \(A=16 e^{-0.0001211}\) describes the amount of carbon-I4 present after t years. Use this model to solve Exercises \(15-16 .\) How many grams of carbon-14 will be present in \(11,430\) years?

Solve each exponential equation by expressing each side as a power of the same base and then equating exponents. $$7^{\frac{x-2}{6}}=\sqrt{7}$$

Write each equation in its equivalent logarithmic form. $$15^{2}=x$$

Graph each function by making a table of coordinates. If applicable, use a graphing utility to confirm your hand-drawn graph. $$h(x)=\left(\frac{1}{3}\right)^{x}$$

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

The half-life of the radioactive element krypton-91 is 10 seconds. If 16 grams of krypton-91 are initially present, how many grams are present after 10 seconds? 20 seconds? 30 seconds? 40 seconds? 50 seconds?

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

Write each equation in its equivalent logarithmic form. $$b^{3}=1000$$

Graph each function by making a table of coordinates. If applicable, use a graphing utility to confirm your hand-drawn graph. $$f(x)=(0.6)^{x}$$

Use properties of logarithms to expand logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. \(\log N^{-6}\)

The half-life of the radioactive element plutonium-239 is \(25,000\) years. If 16 grams of plutonium- 239 are initially present, how many grams are present after \(25,000\) years? \(50,000\) years? \(75,000\) years? \(100,000\) years? \(125,000\) years?

Solve each exponential equation by expressing each side as a power of the same base and then equating exponents. $$9^{x}=\frac{1}{\sqrt[3]{3}}$$

Write each equation in its equivalent logarithmic form. $$b^{3}=343$$

Graph each function by making a table of coordinates. If applicable, use a graphing utility to confirm your hand-drawn graph. $$f(x)=(0.8)^{x}$$

Use properties of logarithms to expand logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. \(\log M^{-8}\)

Solve each exponential equation by expressing each side as a power of the same base and then equating exponents. $$8^{x+3}=16^{x-1}$$

Write each equation in its equivalent logarithmic form. $$7^{y}=200$$

Use properties of logarithms to expand logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. \(\ln \sqrt[5]{x}\)

Use the exponential decay model for carbon- \(14, A=A_{0} e^{-0.000121 t}\) to solve Exercises \(19-20\) Skeletons were found at a construction site in San Francisco in \(1989 .\) The skeletons contained \(88 \%\) of the expected amount of carbon-14 found in a living person. In \(1989,\) how old were the skeletons?

Solve each exponential equation by expressing each side as a power of the same base and then equating exponents. $$8^{1-x}=4^{x+2}$$

Write each equation in its equivalent logarithmic form. $$8^{y}=300$$

Use properties of logarithms to expand logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. \(\ln \sqrt[7]{x}\)

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

Evaluate each expression without using a calculator. $$\log _{4} 16$$

Use properties of logarithms to expand logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. \(\log _{b}\left(x^{2} y\right)\)

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

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

Use properties of logarithms to expand logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. \(\log _{b}\left(x y^{3}\right)\)

Solve each exponential equation. 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. $$10^{x}=3.91$$

Evaluate each expression without using a calculator. $$\log _{2} 64$$

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

Solve each exponential equation. 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. $$10^{x}=8.07$$

Evaluate each expression without using a calculator. $$\log _{3} 27$$

Use properties of logarithms to expand logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. \(\log _{5}\left(\frac{\sqrt{x}}{25}\right)\)

Solve each exponential equation. 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. $$e^{x}=5.7$$

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. $$g(x)=2^{x+1}$$

Evaluate each expression without using a calculator. $$\log _{5} \frac{1}{5}$$

Use properties of logarithms to expand logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. \(\log _{6}\left(\frac{36}{\sqrt{x+1}}\right)\)

Solve each exponential equation. 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. $$e^{x}=0.83$$

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. $$g(x)=2^{x+2}$$

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

Use properties of logarithms to expand logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. \(\log _{8}\left(\frac{64}{\sqrt{x+1}}\right)\)

Solve each exponential equation. 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. $$5^{x}=17$$

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. $$g(x)=2^{x}-1$$