91Ó°ÊÓ

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?) Uranium 235 , used for nuclear power, initial amount \(1 \mathrm{~kg}\) grams, half-life 704 million years.

Evaluate the expression. \(\log _{6}\left(\frac{1}{36}\right)\)

Solve the equation analytically. $$ \log (x)-\log (2)=\log (x+8)-\log (x+2) $$

Solve the equation analytically. $$ \ln (\ln (x))=3 $$

Under optimal conditions, the growth of a certain strain of \(E\). Coli is modeled by the Law of Uninhibited Growth \(N(t)=N_{0} e^{k t}\) where \(N_{0}\) is the initial number of bacteria and \(t\) is the elapsed time, measured in minutes. From numerous experiments, it has been determined that the doubling time of this organism is 20 minutes. Suppose 1000 bacteria are present initially. (a) Find the growth constant \(k\). Round your answer to four decimal places. (b) Find a function which gives the number of bacteria \(N(t)\) after \(t\) minutes. (c) How long until there are 9000 bacteria? Round your answer to the nearest minute.

Solve the equation analytically. $$ (\log (x))^{2}=2 \log (x)+15 $$

Evaluate the expression. \(\log _{4}(8)\)

Evaluate the expression. \(\log \left(\sqrt[9]{10^{11}}\right)\)

Use the appropriate change of base formula to approximate the logarithm. $$ \log _{\frac{3}{5}}(1000) $$

Prove the Quotient Rule and Power Rule for Logarithms.