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Using Properties of Logarithms In Exercises \(15-20\) , use the properties of logarithms to rewrite and simplify the logarithmic expression. $$\ln \frac{6}{e^{2}}$$

Bacteria Growth The number of bacteria in a culture is increasing according to the law of exponential growth. The initial population is 250 bacteria, and the population after 10 hours is double the population after 1 hour. How many bacteria will there be after 6 hours?

Solve the exponential equation algebraically. Approximate the result to three decimal places. \(\left(1+\frac{0.065}{365}\right)^{365 t}=4\)

Expanding a Logarithmic Expression In Exercises \(37-58\) , use the properties of logarithms to expand the expression as a sum, difference, and or constant multiple of logarithms. (Assume all variables are positive.) $$\ln x y z^{2}$$

Due to the installation of noise suppression materials, the noise level in an auditorium decreased from 93 to 80 decibels. Find the percent decrease in the intensity level of the noise as a result of the installation of these materials.

Expanding a Logarithmic Expression In Exercises \(37-58\) , use the properties of logarithms to expand the expression as a sum, difference, and or constant multiple of logarithms. (Assume all variables are positive.) $$\log _{2} \frac{\sqrt{a}-1}{9}, a>1$$

Due to the installation of a muffler, the noise level of an engine decreased from 88 to 72 decibels. Find the percent decrease in the intensity level of the noise as a result of the installation of the muffler.

Forensics At 8: 30 A.M., a coroner went to the home of a person who had died during the night. In order to estimate the time of death, the coroner took the person's temperature twice. At 9: 00 A.M. the temperature was \(85.7^{\circ} \mathrm{F},\) and at 11: 00 A.M. the temperature was \(82.8^{\circ} \mathrm{F} .\) From these two temperatures, the coroner was able to determine that the time elapsed since death and the body temperature were related by the formula \(t=-10 \ln \frac{T-70}{98.6-70}\) where \(t\) is the time in hours elapsed since the person died and \(T\) is the temperature (in degrees Fahrenheit) of the person's body. (This formula comes from a general cooling principle called Newton's Law of Cooling. It uses the assumptions that the person had a normal body temperature of \(98.6^{\circ} \mathrm{F}\) at death and that the room temperature was a constant \(70^{\circ} \mathrm{F}\).) Use the formula to estimate the time of death of the person.

Think About It For how many integers between 1 and 20 can you approximate natural logarithms, given the values \(\ln 2 \approx 0.6931,\) ln \(3 \approx 1.0986,\) and ln 5\(\approx 1.6094 ?\) Approximate these logarithms (do not use a calculator).