91Ó°ÊÓ

Write each equation in logarithmic form. $$27^{\frac{-2}{3}}=\frac{1}{9}$$

Ordinary annuities: If a periodic payment \(\mathcal{P}\) is deposited \(n\) times per year, with annual interest rate \(r\) also compounded \(n\) times per year for \(t\) years, the future value of the account is given by \(A=\frac{P\left((1+R)^{mt}\right)-1}{R}\) where \(R=\frac{r}{n}\) (if the rate is \(9 \%\) compounded monthly, \(\left.R=\frac{0.09}{12}=0.0075\right)\) What quarterly investment amount is required to ensure that Larry can save \(\$ 4700\) in 4 yr at an annual rate of \(8.5 \%\) compounded quarterly?

Determine the value of each logarithm without using a calculator. $$\log _{81} 9$$

Use the properties of logarithms to write the following expressions as a sum or difference of simple logarithmic terms. $$\ln (x \sqrt[4]{y})$$

Use a calculator to evaluate each expression, rounded to four decimal places. $$\ln 0.75$$

The radioactive element iodine- 131 has a half-life of 8 days and is often used to help diagnose patients with thyroid problems. If a certain thyroid procedure requires \(0.5 \mathrm{g}\) and is scheduled to take place in 3 days, what is the minimum amount that must be on hand now (to the nearest hundredth of a gram)?

Solve each exponential equation and check your answer by substituting into the original equation. $$8^{x+2}=32$$

The radioactive element sodium-24 has a half-life of 15 hr and is used to help locate obstructions in blood flow. If the procedure requires \(0.75 \mathrm{g}\) and is scheduled to take place in 2 days \((48 \mathrm{hr}),\) what minimum amount must be on hand now (to the nearest hundredth of a \(\operatorname{gram})^{2}\)

The radioactive element americium-241 has a half-life of 432 yr and although extremely small amounts are used (about \(0.0002 \mathrm{g}),\) it is the most vital component of standard household smoke detectors. How many years will it take a 10 -g mass of americium- 241 to decay to \(2.7 \mathrm{g} ?\)

The growth of a bacteria population: \(P(t)=1000 \cdot 3^{t}\) If the initial population of a common bacterium is 1000 and the population triples every day, its population is given by the formula shown, where \(P(t)\) is the total population after \(t\) days. (a) Find the total population \(12 \mathrm{hr}, 1\) day, \(1 \frac{1}{2}\) days, and 2 days later. (b) Do the outputs show the population is tripling every \(24 \mathrm{hr}\) ( 1 day)? (c) Explain why this is an increasing function. (d) Graph the function using an appropriate scale.