91Ó°ÊÓ

Determine whether the statement is true or false. Justify your answer. The exponential model \(y=a e^{b x}\) represents a growth model when \(b>0.\)

Carbon 14 ( \(^{14} \mathrm{C}\) ) dating assumes that the carbon dioxide on Earth today has the same radioactive content as it did centuries ago. If this is true, then the amount of \(^{14} \mathrm{C}\) absorbed by a tree that grew several centuries ago should be the same as the amount of \(^{14} \mathrm{C}\) absorbed by a tree growing today. \(\mathrm{A}\) piece of ancient charcoal contains only \(15 \%\) as much radioactive carbon as a piece of modern charcoal. How long ago was the tree burned to make the ancient charcoal, given that the half-life of \(^{14} \mathrm{C}\) is about 5700 years?

Use the properties of logarithms to simplify the expression. $$3 \log _{2} \frac{1}{2}$$

Solve the logarithmic equation. $$\ln x=-9$$

Use the properties of logarithms to rewrite and simplify the logarithmic expression.$$\log _{2} 4^{2} \cdot 3^{4}$$.

Use a graphing utility to construct a table of values for the function. Then sketch the graph of the function. Identify any asymptotes of the graph. $$f(x)=6^{x}$$

Solve the logarithmic equation. $$\ln x=-14$$

The half-life of radioactive radium \(\left(^{226} \mathrm{Ra}\right)\) is 1600 years. What percent of a present amount of radioactive radium will remain after 100 years?

Use the properties of logarithms to simplify the expression. $$\frac{1}{4} \log _{4} 16$$

Use the properties of logarithms to rewrite and simplify the logarithmic expression.$$\log _{3} 9^{2} \cdot 2^{4}$$.