Solve the equation $3^{2x} - 30 = 3^x$.

1. Problem Description

Solve the equation $3^{2x} - 30 = 3^x$.

2. Solution Steps

Let $y = 3^x$. Then $3^{2x} = (3^x)^2 = y^2$.

The given equation $3^{2x} - 30 = 3^x$ can be rewritten as $y^2 - 30 = y$.

Rearranging the terms, we have a quadratic equation: $y^2 - y - 30 = 0$.

We can factor this equation as $(y - 6)(y + 5) = 0$.

So, $y = 6$ or $y = -5$.

Since $y = 3^x$, we have $3^x = 6$ or $3^x = -5$.

However, $3^x$ is always positive, so $3^x = -5$ has no real solution.

For $3^x = 6$, we can take the logarithm base 3 of both sides: $x = \log_3 6$.

Using the change of base formula, we can write $x = \frac{\ln 6}{\ln 3}$.

We can also express the solution as $x = \log_3 6 = \log_3 (2 \cdot 3) = \log_3 2 + \log_3 3 = \log_3 2 + 1$.

3. Final Answer

$x = \log_3 6$