91Ó°ÊÓ

Solve the equation. Write the solution set with the exact values given in terms of common or natural logarithms. Also give approximate solutions to 4 decimal places. \(2^{1-6 x}=7^{3 x+4}\)

Write the equation in logarithmic form. $$ a^{7}=b $$

Technetium- \(99\left({ }^{99 \mathrm{~m}} \mathrm{Tc}\right)\) is a radionuclide used widely in nuclear medicine. \({ }^{99 \mathrm{~m}} \mathrm{Tc}\) is combined with another substance that is readily absorbed by a targeted body organ. Then, special cameras sensitive to the gamma rays emitted by the technetium are used to record pictures of the organ. Suppose that a technician prepares a sample of \(^{99 \mathrm{~m}}\) Tc-pyrophosphate to image the heart of a patient suspected of having had a mild heart attack. a. At noon, the patient is given \(10 \mathrm{mCi}\) (millicuries) of \({ }^{99 \mathrm{~m}} \mathrm{Tc}\). If the half-life of \({ }^{99 \mathrm{~m}} \mathrm{Tc}\) is \(6 \mathrm{hr}\), write a function of the form \(Q(t)=Q_{0} e^{-k t}\) to model the radioactivity level \(Q(t)\) after \(t\) hours. b. At what time will the level of radioactivity reach \(3 \mathrm{mCi} ?\) Round to 1 decimal place.

Solve the equation. Write the solution set with the exact values given in terms of common or natural logarithms. Also give approximate solutions to 4 decimal places. \(11^{1-8 x}=9^{2 x+3}\)

Write the equation in logarithmic form. $$ M^{3}=N $$

Solve the equation. Write the solution set with the exact values given in terms of common or natural logarithms. Also give approximate solutions to 4 decimal places. \(e^{2 x}-9 e^{x}-22=0\)

Fluorodeoxyglucose is a derivative of glucose that contains the radionuclide fluorine- \(18\left({ }^{18} \mathrm{~F}\right) .\) A patient is given a sample of this material containing \(300 \mathrm{MBq}\) of \({ }^{18} \mathrm{~F}\) (a megabecquerel is a unit of radioactivity). The patient then undergoes a PET scan (positron emission tomography) to detect areas of metabolic activity indicative of cancer. After \(174 \mathrm{~min}\), one-third of the original dose remains in the body. a. Write a function of the form \(Q(t)=Q_{0} e^{-k t}\) to model the radioactivity level \(Q(t)\) of fluorine- 18 at a time \(t\) minutes after the initial dose. b. What is the half-life of \({ }^{18} \mathrm{~F}\) ?

Simplify the expression. $$ \log 100,000,000 $$

Simplify the expression. $$ \log _{2} 16 $$

Solve the equation. Write the solution set with the exact values given in terms of common or natural logarithms. Also give approximate solutions to 4 decimal places. \(e^{2 x}-6 e^{x}-16=0\)