91Ó°ÊÓ

Plot the functions \(y=2^{x}, y=e^{x},\) and \(y=3^{x}\) in the same viewing screen. Explain why \(y=e^{x}\) lies between the other two graphs.

Calculate the decibels associated with normal comersation if the intensity is \(I=1 \times 10^{-6} \mathrm{W} / \mathrm{m}^{2}\).

Explain the mistake that is made. Solve the equation: \(4 e^{x}=9\) Solution: Take the natural log of both sides. \(\quad \ln \left(4 e^{x}\right)=\ln 9\) Apply the property of inverses. \(4 x=\ln 9\) \(x=\frac{\ln 9}{4} \approx 0.55\) Solve for \(x\) This is incorrect. What mistake was made?

Use a graphing calculator to plot \(y=\frac{\log x}{\log 2}\) and \(y=\log x-\log 2 .\) Are they the same graph?

Plot \(y_{1}=e^{x}\) and \(y_{2}=1+x+\frac{x^{2}}{2}+\frac{x^{3}}{6}+\frac{x^{4}}{24}\) in the same viewing screen. What do you notice?

Explain the mistake that is made. Solve the equation: \(\log (x)+\log (3)=1\) Solution: Apply the product property ( 5 ). \(\log (3 x)=1\) Exponentiate (base 10 ) \(10^{\log (3 x)}=1\) Apply the properties of inverses. \(3 x=1\) Solve for \(x . \quad \quad x=\frac{1}{3}\) This is incorrect. What mistake was made?

Use a graphing calculator to plot \(y=\log \left(\frac{x}{2}\right)\) and \(y=\log x-\log 2 .\) Are they the same graph?

Plot \(y_{1}=e^{-x}\) and \(y_{2}=1-x+\frac{x^{2}}{2}-\frac{x^{3}}{6}+\frac{x^{4}}{24}\) in the same viewing screen. What do you notice?

Use a graphing calculator to plot \(y=\ln \left(x^{2}\right)\) and \(y=2 \ln x\) Are they the same graph?

Solve the equation: \(\log (x)+\log (x+3)=1\) for \(x\) Solution: Apply the product property (5). \(\quad \log \left(x^{2}+3 x\right)=1\) Exponentiate both sides (base 10 ). \(10^{\log \left(x^{2}+3 x\right)}=10^{1}\) Apply the property of inverses. \(x^{2}+3 x=10\) Factor. \(\quad(x+5)(x-2)=0\) Solve for \(x . \quad \quad x=-5\) and \(x=2\) This is incorrect. What mistake was made?