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Magazine Problem 16
In Exercises \(1-33,\) solve the equation analytically. $$ 500\left(1-e^{2 x}\right)=250 $$
Problem 16
Evaluate the expression. \(\log _{3}(27)\)
Problem 16
Solve the equation analytically. $$ \log _{5}(2 x+1)+\log _{5}(x+2)=1 $$
Problem 16
In Exercises \(16-29,\) use the properties of logarithms to write the expression as a single logarithm. $$ 4 \ln (x)+2 \ln (y) $$
Problem 16
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?) Chromium 51, used to track red blood cells, initial amount 75 milligrams, half-life 27.7 days.
Problem 17
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?) Americium 241, used in smoke detectors, initial amount 0.29 micrograms, half- life 432.7 years.
Problem 17
In Exercises \(1-33,\) solve the equation analytically. $$ 70+90 e^{-0.1 t}=75 $$
Problem 17
Evaluate the expression. \(\log _{6}(216)\)
Problem 17
Solve the equation analytically. $$ \log _{169}(3 x+7)-\log _{169}(5 x-9)=\frac{1}{2} $$
Problem 17
Use the properties of logarithms to write the expression as a single logarithm. $$ \log _{2}(x)+\log _{2}(y)-\log _{2}(z) $$
