Chapter 4

Graph each function by making a table of coordinates. If applicable, use a graphing utility to confirm your hand-drawn graph. \(f(x)=4^{x}\)

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. $$ \log _{5}\left(\frac{125}{y}\right) $$

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. $$e^{1-8 x}=7957$$

Write each equation in its equivalent logarithmic form. $$5^{-3}=\frac{1}{125}$$

Graph each function by making a table of coordinates. If applicable, use a graphing utility to confirm your hand-drawn graph. \(f(x)=5^{x}\)

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. $$ \ln \left(\frac{e^{2}}{5}\right) $$

An artifact originally had 16 grams of carbon- 14 present. The decay model \(A=16 e^{-0.0001211}\) describes the amount of carbonI 4 present, \(A,\) in grams, after \(t\) years. How many grams of carbon-14 will be present after 5715 years?

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. $$e^{5 x-3}-2=10,476$$

Write each equation in its equivalent logarithmic form. $$\sqrt[3]{8}=2$$

Graph each function by making a table of coordinates. If applicable, use a graphing utility to confirm your hand-drawn graph. \(g(x)=\left(\frac{3}{2}\right)^{x}\)

An artifact originally had 16 grams of carbon- 14 present. The decay model \(A=16 e^{-0.0001211}\) describes the amount of carbonI 4 present, \(A,\) in grams, after \(t\) years. How many grams of carbon-14 will be present after \(11,430\) years?

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. $$e^{4 x-5}-7=11,243$$

Graph each function by making a table of coordinates. If applicable, use a graphing utility to confirm your hand-drawn graph. \(g(x)=\left(\frac{4}{3}\right)^{x}\)

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. $$ \log _{b} x^{3} $$

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. $$7^{x+2}=410$$

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. \(h(x)=\left(\frac{1}{2}\right)^{x}\)

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

An artifact originally had 16 grams of carbon- 14 present. The decay model \(A=16 e^{-0.0001211}\) describes the amount of carbonI 4 present, \(A,\) in grams, after \(t\) years. 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 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. $$5^{x-3}=137$$

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. \(h(x)=\left(\frac{1}{3}\right)^{x}\)

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

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. $$7^{0.3 x}=813$$

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. \(f(x)=(0.6)^{x}\)

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

Use the exponential decay model for carbon- 14 \(A=A_{0} e^{-0.000121 t},\) 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 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. $$3^{x / 7}=0.2$$

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. \(f(x)=(0.8)^{x}\)

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

Use the exponential decay model for carbon- 14 \(A=A_{0} e^{-0.000121 t},\) The August 1978 issue of National Geographic described the 1964 find of dinosaur 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 approximately 1.31 billion years. Use the fac after 1.31 billion years a given amount of pota 40 will have decayed to half the original amo show that the decay model for potassium-40 is \(A=A_{0 f}^{-0.524}\) 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.

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. $$5^{2 x+3}=3^{x-1}$$

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

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. $$ \ln \sqrt[2]{x} $$

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. $$7^{2 x-1}=3^{x+2}$$

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

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

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

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. $$e^{2 x}-3 e^{x}+2=0$$

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

Use the exponential growth model, \(A=A_{0} e^{k t},\) to show that the time it takes a population to triple (to grow from \(A_{0}\) to \(3 A_{0}\) ) is given by \(t=\frac{\ln 3}{k}\)Use the formula \(t=\frac{\ln 2}{k}\) that gives the time for a population

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. $$e^{2 x}-2 e^{x}-3=0$$

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

With a growth rate \(k\) to double. Express each answer to the nearest whole year. China is growing at a rate of \(1.1 \%\) per year. How long will it take China to double its population?

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. $$e^{4 x}+5 e^{2 x}-24=0$$

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