To solve this logarithmic equation, we first need to rewrite it in exponential form.

The given equation is:

[tex] log_{x}(x^{2} - 2x + 6) = 2 [/tex]

This can be rewritten as:

[tex] x^{2} - 2x + 6 = x^{2} [/tex]

Subtracting [tex] x^{2} [/tex] from both sides gives:

[tex] -2x + 6 = 0 [/tex]

Solving for x by moving 6 to the other side gives:

[tex] -2x = -6 [/tex]

Dividing by -2 gives:

[tex] x = 3 [/tex]

Therefore, the solution to the equation is [tex] x = 3 [/tex].