91Ó°ÊÓ

Which of the following equations is correct? Justify your answer. (A) \(\log _7 x+2 \log _7 y=\log _7\left(x+y^2\right)\) (B) \(9 \log x-2 \log y=\log \frac{x^9}{y^2}\) (C) \(5 \log _4 x+7 \log _2 y=\log _6 x^5 y^7\) (D) \(\log _9 x-5 \log _9 y=\log _9 \frac{x}{5 y}\)

In Exercises 31-34, use a table of values or a graphing calculator to graph the function. Then identify the domain and range. $$ y=e^{x+1} $$

Rewrite the function in the form \(y=a(1+r)^t\) or \(y=a(1-r)^t\). Then state the growth or decay rate. \(y=a(0.25)^{t / 9}\)

Solve the equation. \(\log _3\left(x^2+9 x+27\right)=2\)

Match the function with the correct transformation of the graph of \(f\). Explain your reasoning. \(y=f(x+2)\)

Match the function with the correct transformation of the graph of \(f\). Explain your reasoning. \(y=2 f(x)\)

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 \(50^{\circ} \mathrm{C}\). $$ \begin{array}{|c|c|c|c|c|c|c|} \hline \boldsymbol{d} & 160 & 90 & 56 & 38 & 29 & 24 \\ \hline \boldsymbol{t} & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline \end{array} $$

Use the change-of-base formula to evaluate the logarithm. $$\log _4 7$$

Solve the equation. Check for extraneous solutions. \(\log _2 x+\log _2(x-2)=3\)

In Exercises 31-34, use a table of values or a graphing calculator to graph the function. Then identify the domain and range. $$ y=2 e^x+1 $$