Chapter 12

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. If \(\log (x+3)=2,\) then \(e^{2}=x+3\)

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. If \(x=\frac{1}{k} \ln y,\) then \(y=e^{k x}\)

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. Examples of exponential equations include \(10^{x}=5.71\) \(e^{x}=0.72,\) and \(x^{10}=5.71\)

Solve each equation in Exercises \(144-146 .\) Check each proposed solution by direct substitution or with a graphing utility. $$\ln (\ln x)=0$$

Solve: $$\sqrt{2 x-1}-\sqrt{x-1}=1$$

Solve: $$\frac{3}{x+1}-\frac{5}{x}=\frac{19}{x^{2}+x}$$

Simplify: \(\left(-2 x^{3} y^{-2}\right)^{-4}\)

The formula \(A=10 e^{-0.003 t}\) models the population of Hungary, \(A\), in millions, \(t\) years after 2006 . a. Find Hungary's population, in millions, for \(2006,2007\), \(2008,\) and \(2009 .\) Round to two decimal places. b. Is Hungary's population increasing or decreasing?

a. Simplify: \(e^{\ln 3}\) b. Use your simplification from part (a) to rewrite \(3^{x}\) in terms of base \(e\)

U.S. soldiers fight Russian troops who have invaded New York City. Incoming missiles from Russian submarines and warships ravage the Manhattan skyline. It's just another scenario for the multi-billion-dollar video games Call of Duty, which have sold more than 100 million games since the franchise's birth in 2003 . The table shows the annual retail sales for Call of Duty video games from 2004 through 2010 . Create a scatter plot for the data. Based on the shape of the scatter plot, would a logarithmic function, an exponential function, or a linear function be the best choice for modeling the data? $$\begin{array}{l|c} \hline \text { Annual Retail Sales for } \text {Call of Duty Games} \\ \hline \text { Year } & \begin{array}{c} \text { Retail Sales } \\ \text { (millions of dollars) } \end{array} \\ \hline 2004 & 56 \\ \hline 2005 & 101 \\ \hline 2006 & 196 \\ \hline 2007 & 352 \\ \hline 2008 & 436 \\ \hline 2009 & 778 \\ \hline 2010 & 980 \\ \hline \end{array}$$