91Ó°ÊÓ

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

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

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

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

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. $$ \log _{3}(7-2 x)=2 $$

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

We list some radioactive isotopes and their associated half-lives. Assume that each decays according to the formula \(A(t)=A_{0} e^{k t}\) where \(A_{0}\) is the initial amount of the material and \(k\) is the decay constant. For each isotope: \- Find the decay constant \(k\). Round your answer to four decimal places. \- Find a function which gives the amount of isotope \(A\) which remains after time \(t\). (Keep the units of \(A\) and \(t\) the same as the given data.) \- Determine how long it takes for \(90 \%\) of the material to decay. Round your answer to two decimal places. (HINT: If \(90 \%\) of the material decays, how much is left?) Phosphorus 32 , used in agriculture, initial amount 2 milligrams, half-life 14 days.

Solve the equation analytically. $$ \log _{5}(2 x+1)+\log _{5}(x+2)=1 $$

Evaluate the expression. \(\log _{6}(216)\)