91Ó°ÊÓ

The number \(y\) of duckweed fronds in a pond after \(t\) days is \(y=a(1230.25)^{t / 16}\), where \(a\) is the initial number of fronds. By what percent does the duckweed increase each day?

Rewrite the function in the form \(y=a(1+r)^t\) or \(y=a(1-r)^t\). Then state the growth or decay rate. \(y=a(2)^{t / 3}\)

In Exercises 27-30, use the properties of exponents to rewrite the function in the form \(y=a(1+r)^t\) or \(y=a(1-r)^t\). Then find the percent rate of change. $$ y=0.5 e^{0.8 t} $$

Which of the following is not equivalent to \(\log _5 \frac{y^4}{3 x}\) ? Justify your answer. (A) \(4 \log _5 y-\log _5 3 x\) (B) \(4 \log _5 y-\log _5 3+\log _5 x\) (C) \(4 \log _5 y-\log _5 3-\log _5 x\) (D) \(\log _5 y^4-\log _5 3-\log _5 x\)

An object at a temperature of \(160^{\circ} \mathrm{C}\) is removed from a furnace and placed in a room at \(20^{\circ} \mathrm{C}\). The table shows the temperatures \(d\) (in degrees Celsius) at selected times \(t\) (in hours) after the object was removed from the furnace. Use a graphing calculator to find a logarithmic model of the form \(t=a+b \ln d\) that represents the data. Estimate how long it takes for the object to cool to \(50^{\circ} \mathrm{C}\). $$ \begin{array}{|c|c|c|c|c|c|c|} \hline \boldsymbol{d} & 160 & 90 & 56 & 38 & 29 & 24 \\ \hline \boldsymbol{t} & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline \end{array} $$

The f-stops on a camera control the amount of light that enters the camera. Let \(s\) be a measure of the amount of light that strikes the film and let \(f\) be the f-stop. The table shows several f-stops on a 35-millimeter camera. Use a graphing calculator to find a logarithmic model of the form \(s=a+b \ln f\) that represents the data. Estimate the amount of light that strikes the film when \(f=5.657\). $$ \begin{array}{|c|c|} \hline \boldsymbol{f} & \boldsymbol{s} \\ \hline 1.414 & 1 \\ 2.000 & 2 \\ 2.828 & 3 \\ 4.000 & 4 \\ 11.314 & 7 \\ \hline \end{array} $$

In Exercises 31-34, use a table of values or a graphing calculator to graph the function. Then identify the domain and range. $$ y=3 e^x-5 $$

The table shows the average weight (in kilograms) of an Atlantic cod that is \(x\) years old from the Gulf of Maine. $$ \begin{array}{|l|c|c|c|c|c|} \hline \text { Age, } \boldsymbol{x} & 1 & 2 & 3 & 4 & 5 \\ \hline \text { Weight, } \boldsymbol{y} & 0.751 & 1.079 & 1.702 & 2.198 & 3.438 \\ \hline \end{array} $$ a. Show that an exponential model fits the data. Then find an exponential model for the data. b. By what percent does the weight of an Atlantic cod increase each year in this period of time? Explain.

Find values of \(a, b, r\), and \(q\) such that \(f(x)=a e^{r x}\) and \(g(x)=b e^{q x}\) are exponential decay functions, but \(\frac{f(x)}{g(x)}\) represents exponential growth.

You deposit \(\$ 5000\) in an account that pays \(2.25 \%\) annual interest. Find the balance after 5 years when the interest is compounded quarterly. (See Example 5.)