91Ó°ÊÓ

Growth of Bacteria. The bacteria Escherichia coli (E. coli) are commonly found in the human bladder. Suppose that 3000 of the bacteria are present at time \(t=0 .\) Then \(t\) minutes later, the number of bacteria present is $$ N(t)=3000(2)^{t / 20} $$ If \(100,000,000\) bacteria accumulate, a bladder infection can occur. If, at 11: 00 A.M., a patient's bladder contains \(25,000 E\) coli bacteria, at what time can infection occur?

For each function, (a) determine whether it is one-to-one; (b) if it is one- to-one, find a formula for the inverse. \(g(x)=(x+7)^{3}\)

Solve. Where appropriate, include approximations to three decimal places. If no solution exists, state this. $$ \log _{6}(x+3)+\log _{6}(x+2)=\log _{6} 20 $$

Express as an equivalent expression that is a single logarithm and, if possible, simplify. $$\log _{a} 2 x+3\left(\log _{a} x-\log _{a} y\right)$$

Show that for exponential growth at rate \(k,\) the doubling time \(T\) is given by \(T=\frac{\ln 2}{k}\).

Express as an equivalent expression that is a single logarithm and, if possible, simplify. $$\log _{a}\left(x^{2}-9\right)-\log _{a}(x+3)$$

Solve. Where appropriate, include approximations to three decimal places. If no solution exists, state this. $$ \log _{5}(x+4)+\log _{5}(x-4)=\log _{5} 20 $$

For each function, (a) determine whether it is one-to-one; (b) if it is one- to-one, find a formula for the inverse. \(f(x)=\sqrt{x}\)

Rewrite each of the following as an equivalent exponential equation. Do not solve. $$ \log _{e} m=8 $$

Rewrite each of the following as an equivalent exponential equation. Do not solve. $$ \log _{b} n=23 $$