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.