The problem asks us to solve the exponential equation $3^{2-x} = \frac{1}{27}$ for $x$, without using logarithms.

1. Problem Description

The problem asks us to solve the exponential equation $3^{2-x} = \frac{1}{27}$ for $x$, without using logarithms.

2. Solution Steps

We need to express both sides of the equation with the same base.

Since $27 = 3^3$, we can write $\frac{1}{27}$ as $\frac{1}{3^3}$.

Also, $\frac{1}{3^3}$ can be written as $3^{-3}$.

Thus, the equation becomes

$3^{2-x} = 3^{-3}$.

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

$2 - x = -3$.

Now, we can solve for $x$:

$2 - x = -3$

$-x = -3 - 2$

$-x = -5$

$x = 5$

3. Final Answer

$x = 5$