College Algebra, Chapter 5, Review Exercise, Section Review Exercise, Problem 64

Evaluate the equaiton $3^{2x} - 3^x - 6 = 0$. Find the exact solution, otherwise use a calculator.

$
\begin{equation}
\begin{aligned}
3^{2x} - 3^x - 6 &= 0 \\
\\
(3^x - 3)(3^x + 2) &= 0 && \text{Factor}
\end{aligned}
\end{equation}
$

Solve for $x$

$
\begin{array}{rll|rrl}
3^x - 3 &= 0 & && 3^x + 2 &= 0\\
\\
3^x &= 3 & \text{Add 3} && 3^x &= -2 & \text{Subtract 2}\\
\\
\ln 3^x &= \ln 3 & \text{Take ln of each side} && \ln3^x &= \ln(-2) & \text{Take ln of each side}\\
\\
x \ln 3 &= \ln 3 & \text{Properties of ln} && x \ln 3 &= \ln(-2) & \text{Properties of ln}\\
\\
x &= \displaystyle \frac{\ln3}{\ln3} & \text{Divide by ln 3} && x &= \displaystyle \frac{\ln(-2)}{\ln(3)} & \text{Divide by ln 3}\\
\\
x &= 1
\end{array}
$

The only solution of the given equation is $x = 1$, since $x$ can't have a negative value.