Chapter 6

For each of the scenarios given in Exercises \(1-6\), \- Find the amount \(A\) in the account as a function of the term of the investment \(t\) in years. \- Determine how much is in the account after 5 years, 10 years, 30 years and 35 years. Round your answers to the nearest cent. \- Determine how long will it take for the initial investment to double. Round your answer to the nearest year. \- Find and interpret the average rate of change of the amount in the account from the end of the fourth year to the end of the fifth year, and from the end of the thirty-fourth year to the end of the thirty-fifth year. Round your answer to two decimal places. $$\$ 500$$ is invested in an account which offers \(0.75 \%\), compounded monthly.

Solve the equation analytically. $$ \log (3 x-1)=\log (4-x) $$

In Exercises \(1-33,\) solve the equation analytically. $$ 2^{d x}=8 $$

In Exercises 1 - 15 , expand the given logarithm and simplify. Assume when necessary that all quantities represent positive real numbers. $$ \ln \left(x^{3} y^{2}\right) $$

Use the property: \(b^{a}=c\) if and only if \(\log _{b}(c)=a\) from Theorem 6.2 to rewrite the given equation in the other form. That is, rewrite the exponential equations as logarithmic equations and rewrite the logarithmic equations as exponential equations. \(2^{3}=8\)

For each of the scenarios given in Exercises \(1-6\), \- Find the amount \(A\) in the account as a function of the term of the investment \(t\) in years. \- Determine how much is in the account after 5 years, 10 years, 30 years and 35 years. Round your answers to the nearest cent. \- Determine how long will it take for the initial investment to double. Round your answer to the nearest year. \- Find and interpret the average rate of change of the amount in the account from the end of the fourth year to the end of the fifth year, and from the end of the thirty-fourth year to the end of the thirty-fifth year. Round your answer to two decimal places. $$\$ 500$$ is invested in an account which offers \(0.75 \%\), compounded continuously.

Solve the equation analytically. $$ \log _{2}\left(x^{3}\right)=\log _{2}(x) $$

In Exercises \(1-33,\) solve the equation analytically. $$ 3^{(x-1)}=27 $$

Expand the given logarithm and simplify. Assume when necessary that all quantities represent positive real numbers. $$ \log _{2}\left(\frac{128}{x^{2}+4}\right) $$

Use the property: \(b^{a}=c\) if and only if \(\log _{b}(c)=a\) from Theorem 6.2 to rewrite the given equation in the other form. That is, rewrite the exponential equations as logarithmic equations and rewrite the logarithmic equations as exponential equations. \(5^{-3}=\frac{1}{125}\)

For each of the scenarios given in Exercises \(1-6\), \- Find the amount \(A\) in the account as a function of the term of the investment \(t\) in years. \- Determine how much is in the account after 5 years, 10 years, 30 years and 35 years. Round your answers to the nearest cent. \- Determine how long will it take for the initial investment to double. Round your answer to the nearest year. \- Find and interpret the average rate of change of the amount in the account from the end of the fourth year to the end of the fifth year, and from the end of the thirty-fourth year to the end of the thirty-fifth year. Round your answer to two decimal places. $$\$ 1000$$ is invested in an account which offers \(1.25 \%\), compounded monthly.

Solve the equation analytically. $$ \ln \left(8-x^{2}\right)=\ln (2-x) $$

In Exercises \(1-33,\) solve the equation analytically. $$ 5^{2 x-1}=125 $$

Expand the given logarithm and simplify. Assume when necessary that all quantities represent positive real numbers. $$ \log _{5}\left(\frac{z}{25}\right)^{3} $$

Use the property: \(b^{a}=c\) if and only if \(\log _{b}(c)=a\) from Theorem 6.2 to rewrite the given equation in the other form. That is, rewrite the exponential equations as logarithmic equations and rewrite the logarithmic equations as exponential equations. \(4^{5 / 2}=32\)

For each of the scenarios given in Exercises \(1-6\), \- Find the amount \(A\) in the account as a function of the term of the investment \(t\) in years. \- Determine how much is in the account after 5 years, 10 years, 30 years and 35 years. Round your answers to the nearest cent. \- Determine how long will it take for the initial investment to double. Round your answer to the nearest year. \- Find and interpret the average rate of change of the amount in the account from the end of the fourth year to the end of the fifth year, and from the end of the thirty-fourth year to the end of the thirty-fifth year. Round your answer to two decimal places. $$\$ 1000$$ is invested in an account which offers \(1.25 \%\), compounded continuously.

Solve the equation analytically. $$ \log _{5}\left(18-x^{2}\right)=\log _{5}(6-x) $$

In Exercises \(1-33,\) solve the equation analytically. $$ 4^{2 x}=\frac{1}{5} $$

Expand the given logarithm and simplify. Assume when necessary that all quantities represent positive real numbers. $$ \log \left(1.23 \times 10^{37}\right) $$

Use the property: \(b^{a}=c\) if and only if \(\log _{b}(c)=a\) from Theorem 6.2 to rewrite the given equation in the other form. That is, rewrite the exponential equations as logarithmic equations and rewrite the logarithmic equations as exponential equations. \(\left(\frac{1}{3}\right)^{-2}=9\)

For each of the scenarios given in Exercises \(1-6\), \- Find the amount \(A\) in the account as a function of the term of the investment \(t\) in years. \- Determine how much is in the account after 5 years, 10 years, 30 years and 35 years. Round your answers to the nearest cent. \- Determine how long will it take for the initial investment to double. Round your answer to the nearest year. \- Find and interpret the average rate of change of the amount in the account from the end of the fourth year to the end of the fifth year, and from the end of the thirty-fourth year to the end of the thirty-fifth year. Round your answer to two decimal places. $$\$ 5000$$ is invested in an account which offers \(2.125 \%\), compounded monthly.

Solve the equation analytically. $$ \log _{3}(7-2 x)=2 $$

In Exercises \(1-33,\) solve the equation analytically. $$ 8^{x}=\frac{1}{128} $$

Expand the given logarithm and simplify. Assume when necessary that all quantities represent positive real numbers. $$ \ln \left(\frac{\sqrt{z}}{x y}\right) $$

Use the property: \(b^{a}=c\) if and only if \(\log _{b}(c)=a\) from Theorem 6.2 to rewrite the given equation in the other form. That is, rewrite the exponential equations as logarithmic equations and rewrite the logarithmic equations as exponential equations. \(\left(\frac{4}{25}\right)^{-1 / 2}=\frac{5}{2}\)

For each of the scenarios given in Exercises \(1-6\), \- Find the amount \(A\) in the account as a function of the term of the investment \(t\) in years. \- Determine how much is in the account after 5 years, 10 years, 30 years and 35 years. Round your answers to the nearest cent. \- Determine how long will it take for the initial investment to double. Round your answer to the nearest year. \- Find and interpret the average rate of change of the amount in the account from the end of the fourth year to the end of the fifth year, and from the end of the thirty-fourth year to the end of the thirty-fifth year. Round your answer to two decimal places. $$\$ 5000\( is invested in an account which offers \)2.125 \%$, compounded continuously.

Solve the equation analytically. $$ \log _{\frac{1}{2}}(2 x-1)=-3 $$

In Exercises \(1-33,\) solve the equation analytically. $$ 2^{\left(x^{3}-x\right)}=1 $$

Expand the given logarithm and simplify. Assume when necessary that all quantities represent positive real numbers. $$ \log _{5}\left(x^{2}-25\right) $$

Use the property: \(b^{a}=c\) if and only if \(\log _{b}(c)=a\) from Theorem 6.2 to rewrite the given equation in the other form. That is, rewrite the exponential equations as logarithmic equations and rewrite the logarithmic equations as exponential equations. \(10^{-3}=0.001\)

Solve the equation analytically. $$ \ln \left(x^{2}-99\right)=0 $$

In Exercises \(1-33,\) solve the equation analytically. $$ 3^{7 x}=81^{4-2 x} $$

Expand the given logarithm and simplify. Assume when necessary that all quantities represent positive real numbers. $$ \log _{\sqrt{2}}\left(4 x^{3}\right) $$

Use the property: \(b^{a}=c\) if and only if \(\log _{b}(c)=a\) from Theorem 6.2 to rewrite the given equation in the other form. That is, rewrite the exponential equations as logarithmic equations and rewrite the logarithmic equations as exponential equations. \(e^{0}=1\)

How much money needs to be invested now to obtain \(\$ 2000\) in 3 years if the interest rate in a savings account is \(0.25 \%\), compounded continuously? Round your answer to the nearest cent.

Solve the equation analytically. $$ \log \left(x^{2}-3 x\right)=1 $$

In Exercises \(1-33,\) solve the equation analytically. $$ 9 \cdot 3^{7 x}=\left(\frac{1}{9}\right)^{2 x} $$

Expand the given logarithm and simplify. Assume when necessary that all quantities represent positive real numbers. $$ \log _{\frac{1}{3}}\left(9 x\left(y^{3}-8\right)\right) $$

Use the property: \(b^{a}=c\) if and only if \(\log _{b}(c)=a\) from Theorem 6.2 to rewrite the given equation in the other form. That is, rewrite the exponential equations as logarithmic equations and rewrite the logarithmic equations as exponential equations. \(\log _{5}(25)=2\)

How much money needs to be invested now to obtain $$\$ 5000$$ in 10 years if the interest rate in a CD is \(2.25 \%\), compounded monthly? Round your answer to the nearest cent.

Solve the equation analytically. $$ \log _{125}\left(\frac{3 x-2}{2 x+3}\right)=\frac{1}{3} $$

In Exercises \(1-33,\) solve the equation analytically. $$ 3^{2 x}=5 $$

Expand the given logarithm and simplify. Assume when necessary that all quantities represent positive real numbers. $$ \log \left(1000 x^{3} y^{5}\right) $$

Use the property: \(b^{a}=c\) if and only if \(\log _{b}(c)=a\) from Theorem 6.2 to rewrite the given equation in the other form. That is, rewrite the exponential equations as logarithmic equations and rewrite the logarithmic equations as exponential equations. \(\log _{25}(5)=\frac{1}{2}\)

Solve the equation analytically. $$ \log \left(\frac{x}{10^{-3}}\right)=4.7 $$

In Exercises \(1-33,\) solve the equation analytically. $$ 5^{-x}=2 $$

Expand the given logarithm and simplify. Assume when necessary that all quantities represent positive real numbers. $$ \log _{3}\left(\frac{x^{2}}{81 y^{4}}\right) $$

Use the property: \(b^{a}=c\) if and only if \(\log _{b}(c)=a\) from Theorem 6.2 to rewrite the given equation in the other form. That is, rewrite the exponential equations as logarithmic equations and rewrite the logarithmic equations as exponential equations. \(\log _{3}\left(\frac{1}{81}\right)=-4\)

Solve the equation analytically. $$ -\log (x)=5.4 $$

Expand the given logarithm and simplify. Assume when necessary that all quantities represent positive real numbers. $$ \ln \left(\sqrt[4]{\frac{x y}{e z}}\right) $$