Chapter 3: Chapter 3

a. Given that \(2^{x}=e^{k x}\), find \(k\). b. Show that, in general, if \(b\) is a nonnegative real number, then any equation of the form \(y=b^{x}\) may be written in the form \(y=e^{k x}\), for some real number \(k\).

Use the definition of a logarithm to prove a. \(\log _{b} m n=\log _{b} m+\log _{b} n\) b. \(\log _{b} \frac{m}{n}=\log _{b} m-\log _{b} n\)

Use the definition of a logarithm to prove $$ \log _{b} m^{n}=n \log _{b} m $$

Use the definition of a logarithm to prove a. \(\log _{b} 1=0\) b. \(\log _{b} b=1\)

Express each equation in logarithmic form. $$81^{3 / 4}=27$$

Simplify the expression. a. \(\left(2 x^{3}\right)\left(-4 x^{-2}\right)\) b. \(\left(4 x^{-2}\right)\left(-3 x^{5}\right)\)

Simplify the expression. a. \(\frac{6 a^{-5}}{3 a^{-3}}\) b. \(\frac{4 b^{-4}}{12 b^{-6}}\)

Express each equation in logarithmic form. $$10^{-3}=0.001$$

Phosphorus 32 (P-32) has a half-life of \(14.2\) days. If \(100 \mathrm{~g}\) of this substance are present initially, find the amount present after \(t\) days. What amount will be left after \(7.1\) days?