Certainly, let's solve the equation step-by-step.
The given equation is:
[tex]\[ \frac{1}{36} = 6^{x-4} \][/tex]
Step 1: Rewrite [tex]\(\frac{1}{36}\)[/tex] using base 6.
We know that:
[tex]\[ \frac{1}{36} = \frac{1}{6^2} = 6^{-2} \][/tex]
So, the equation becomes:
[tex]\[ 6^{-2} = 6^{x-4} \][/tex]
Step 2: Equate the exponents.
Since the bases on both sides of the equation are the same (both are base 6), we can set the exponents equal to each other:
[tex]\[ -2 = x - 4 \][/tex]
Step 3: Solve for [tex]\(x\)[/tex].
To isolate [tex]\(x\)[/tex], we add 4 to both sides of the equation:
[tex]\[ -2 + 4 = x - 4 + 4 \][/tex]
[tex]\[ 2 = x \][/tex]
So, the value of [tex]\(x\)[/tex] is:
[tex]\[ \boxed{2} \][/tex]
