42. solve the equation. Write the solution set with the exact same values given in terms of common or natural logarithms. Also give approximate solutions to four decimal places

80=320e^-.5t

56. Solve the equation. Write the solution set with the exact solutions. Also give approximate solutions to four decimal places if necessary.

5log₃(7w+1)+3=13

ln x +ln(x-3)=ln(5x-7)

Use the model A=Pe^rt of A=P(1+r/n) ^nt where A is the future value of P dollars invested at interest rate r compounded continuously or n times per year for t years

66. $20,000 is invested at 3.5% interest compounded monthly. How long will it take for the investment to double? Round to the nearest tenth of a year

72. Caffeine occurs naturally in a variety of food products such as coffee, tea and chocolate. The kidneys filter the blood and remove caffeine and other drugs through urine. The biological half-life of caffeine is approximately 6 hours. If one cup of coffee has 80mg of caffeine, then the amount of caffeine C (in mg) remaining after t hours is given by C=80(2)^-t/6.

A-How long will it take for the amount of caffeine to drop below 60mg? Round to 1 decimal place

B-Laura has trouble sleeping if she has more than 30mg of caffeine in her bloodstream. How many hours after drinking a cup of coffee would Laura have to wait so that the coffee would not disrupt her sleep? Round to 1 decimal place.

86. Use the formula pH=-log[H⁺]. The variable pH represents the level of acidity or alkalinity of a liquid on the pH scale and H+ is the concentration of hydronium ions in the solution. Determine the value of H+ (in mol/L) for the following liquids, given their pH values.

On August 31, 1854, an epidemic of cholera was discovered in London, England, resulting from a contaminated community water pump. By the end of September, more than 600 citizens who drank water from the pump had died. The cumulative number of deaths D(t) at a time t days after August 31 is given by D(t)=91+160 ln(t+1)

A-Determine the cumulative number of deaths by September 15. Round to the nearest whole unit.

B-Approximately how many days after August 31 did the cumulative number of deaths reach 600?