91Ó°ÊÓ

Rewrite each of the following functions using base \(e\). a. \(N=10(1.045)^{t}\) c. \(P=500(2.10)^{x}\) b. \(Q=\left(5 \cdot 10^{-7}\right) \cdot(0.072)^{A}\)

Determine the rule(s) of logarithms that were used to expand each expression. a. \(\ln 15=\ln 3+\ln 5\) b. \(\ln 15=\ln 30-\ln 2\) c. \(\ln 49=2 \ln 7\) d. \(\ln 25 z^{3}=2 \ln 5+3 \ln z\) e. \(\ln 5 x^{4}=\ln 5+4 \ln x\) f. \(\ln \left(\frac{125}{3 x}\right)=3 \ln 5-(\ln 3+\ln x)\)

Identify each of the following functions as representing growth or decay: a. \(Q=N e^{-0.029 t}\) c. \(f(t)=375 e^{0.055 t}\) b. \(h(r)=100(0.87)^{r}\)

Use rules of logarithms to contract to a single logarithm. Use a calculator to verify your answer. a. \(2 \ln 3+4 \ln 2\) c. \(2(\ln 4-\ln 3)\) b. \(3 \ln 7-5 \ln 3\) d. \(-4 \ln 3+\ln 3\)

Expand each logarithm using only the numbers \(2,3, \log 2,\) and \(\log 3\). a. \(\log 9\) b. \(\log 18\) c. \(\log 54\)

Solve for \(x\) by changing to exponential form. Round your answer to three decimal places. a. \(\ln 3 x=1\) b. \(3 \ln x=5\) c. \(\ln 3+\ln x=1.5\)

An investment pays \(6 \%\) compounded four times a year. a. What is the annual growth factor? b. What is the annual growth rate? c. Develop a formula to represent the total value of the investment after each compounding period. d. If you invest \(\$ 2000\) for a child's college fund, how much will it total after 15 years? e. For how many years would you have to invest to increase the total to \(\$ 5000 ?\)

The effective annual interest rate on an account compounded continuously is \(3.38 \%\). Estimate the nominal interest rate.

Solve the following for \(r\). a. \(0.9=e^{r}\) b. \(2=e^{r}\) c. \(0.75=e^{r}\)

The half-life of bismuth-214 is about 20 minutes. a. Construct a function to model the decay of bismuth- 214 over time. Be sure to specify your variables and their units. b. For any given sample of bismuth- 214 , how much is left after I hour? c. How long will it take to reduce the sample to \(25 \%\) of its original size? d. How long will it take to reduce the sample to \(10 \%\) of its original size?