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. $$ \log _{b} x^{7} $$

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

In Exercises \(11-18\), graph each function by making a table of coordinates. If applicable, use a graphing utility to confirm your hand-drawn graph. $$ h(x)-\left(\frac{1}{3}\right)^{x} $$

The half-life of the radioactive element krypton-91 is 10 scoonds. 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?

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. $$ \log N^{-6} $$

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

In Exercises \(11-18\), graph each function by making a table of coordinates. If applicable, use a graphing utility to confirm your hand-drawn graph. $$ f(x)-(0.6)^{x} $$

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?

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. $$ \log M^{-8} $$

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

In Exercises \(11-18\), 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 the exponential decay model for carbon- \(14, A-A_{1} e^{-0,0001211}\) to solve Exercises \(19-20 .\) Prehistoric cave paintings were discovered in a cave in France. The paint contained \(15 \%\) of the original carbon- 14 . Estimate the age of the paintings.

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. $$ \ln \sqrt[5]{x} $$

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

Use the exponential decay model for carbon- \(14, A-A_{1} e^{-0,0001211}\) 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?

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. $$ \ln \sqrt[7]{x} $$

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

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

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. $$ \log _{b}\left(x^{2} y\right) $$

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

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

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. $$ 10^{x}-3.91 $$

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

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

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. $$ 10^{x}-8.07 $$

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

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. $$ \log _{5}\left(\frac{\sqrt{x}}{25}\right) $$

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. $$ e^{x}-5.7 $$

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

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. $$ \log _{6}\left(\frac{36}{\sqrt{x+1}}\right) $$

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. $$ e^{x}-0.83 $$

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

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. $$ \log _{x}\left(\frac{64}{\sqrt{x+1}}\right) $$

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. $$ 5^{x}-17 $$

The August 1978 issue of National Geographic described the 1964 find of bones of a newly discovered dinosaur weighing 170 pounds, measuring 9 feet, with a 6 -inch claw on one toe of each hind foot. The age of the dinosaur was estimated using potassium-40 dating of rocks surrounding the bones. a. Potassium-40 decays exponentially with a half-life of approximately 1.31 billion years. Use the fact that after 1.31 billion years a given amount of potassium- 40 will have decayed to half the original amount to show that the decay model for potassium- 40 is given by \(A-A_{0} e^{-1.52912 t}\) where \(t\) is in billions of years. b. Analysis of the rocks surrounding the dinosaur bones indicated that \(94.5 \%\) of the original amount of potassium- 40 was still present. Let \(A-0.945 A_{0}\) in the model in part (a) and estimate the age of the bones of the dinosaur.

Evaluate each expression without using a calculator. $$ \log _{2} \frac{1}{8} $$

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. $$ \log _{b}\left(\frac{x^{2} y}{z^{2}}\right) $$

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. $$ 19^{x}-143 $$

Use the exponential decay model, \(A=A_{0} e^{k_{1}},\) to solve Exercises \(28-31 .\) Round answers to one decimal place. The half-life of thorium-229 is 7340 years. How long will it take for a sample of this substance to decay to \(20 \%\) of its original amount?

Evaluate each expression without using a calculator. $$\log _{3} \frac{1}{4}$$

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. $$ \log _{b}\left(\frac{x^{3} y}{z^{2}}\right) $$

Use the exponential decay model, \(A=A_{0} e^{k_{1}},\) to solve Exercises \(28-31 .\) Round answers to one decimal place. The half-life of lead is 22 years. How long will it take for a sample of this substance to decay to \(80 \%\) of its original amount?

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

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. $$ \log \sqrt{100 x} $$

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. $$ 9 e^{x}-107 $$

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

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. $$ \ln \sqrt{e x} $$

Evaluate each expression without using a calculator. $$\log _{2} \frac{1}{\sqrt{2}}$$

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. $$ \log \sqrt[3]{\frac{x}{y}} $$

Evaluate each expression without using a calculator. $$\log _{3} \frac{1}{\sqrt{3}}$$