We need to solve the equation $3^{x+4} = (\frac{1}{27})^x$ for $x$.

1. Problem Description

We need to solve the equation $3^{x+4} = (\frac{1}{27})^x$ for $x$.

2. Solution Steps

First, express both sides of the equation using the same base. Since $27 = 3^3$, we can write $\frac{1}{27} = \frac{1}{3^3} = 3^{-3}$. Therefore, $(\frac{1}{27})^x = (3^{-3})^x = 3^{-3x}$.

Thus, the equation becomes:

$3^{x+4} = 3^{-3x}$

Since the bases are equal, we can equate the exponents:

$x+4 = -3x$

Now, solve for $x$. Add $3x$ to both sides:

$x+3x+4 = -3x+3x$

$4x+4 = 0$

Subtract 4 from both sides:

$4x+4-4 = 0-4$

$4x = -4$

Divide both sides by 4:

$\frac{4x}{4} = \frac{-4}{4}$

$x = -1$

3. Final Answer

$x = -1$