91Ó°ÊÓ

Use \(\log _7 4 \approx 0.712\) and \(\log _7 12 \approx 1.277\) to evaluate the logarithm. (See Example 1.) $$\log _7 \frac{1}{4}$$

Write an exponential function \(y=a b^x\) whose graph passes through the given points. \((2,24),(3,144)\)

Solve the equation. \(512^{5 x-1}=\left(\frac{1}{8}\right)^{-4-x}\)

Tell whether the function represents exponential growth or exponential decay. Then graph the function. \(y=(1.2)^x\)

In Exercises 15–22, tell whether the function represents exponential growth or exponential decay. Then graph the function. $$ y=2 e^{-x} $$

The length \(\ell\) (in centimeters) of a scalloped hammerhead shark can be modeled by the function $$ \ell=266-219 e^{-0.05 t} $$ where \(t\) is the age (in years) of the shark. How old is a shark that is 175 centimeters long?

One hundred grams of radium are stored in a container. The amount \(R\) (in grams) of radium present after \(t\) years can be modeled by \(R=100 e^{-0.00043 t}\). After how many years will only 5 grams of radium be present?

A store sells motorized scooters. The table shows the numbers \(y\) of scooters sold during the \(x\) th year that the store has been open. Write a function that models the data. $$ \begin{array}{|c|c|} \hline \boldsymbol{x} & \boldsymbol{y} \\ \hline 1 & 9 \\ 2 & 14 \\ 3 & 19 \\ 4 & 25 \\ 5 & 37 \\ 6 & 53 \\ 7 & 71 \\ \hline \end{array} $$

The table shows the numbers \(y\) of visits to a website during the \(x\) th month. Write a function that models the data. Then use your model to predict the number of visits after 1 year. $$ \begin{array}{|c|c|c|c|c|c|c|c|} \hline \boldsymbol{x} & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \hline \boldsymbol{y} & 22 & 39 & 70 & 126 & 227 & 408 & 735 \\ \hline \end{array} $$

You cook a turkey until the internal temperature reaches \(180^{\circ} \mathrm{F}\). The turkey is placed on the table until the internal temperature reaches \(100^{\circ} \mathrm{F}\) and it can be carved. When the room temperature is \(72^{\circ} \mathrm{F}\), the cooling rate of the turkey is \(r=0.067\). How long do you have to wait until you can carve the turkey?