Domain of $7^{\log_7(x^2-4x+5)}$

The tag says "complex numbers". In complex numbers you do not have any restriction that $x^2 - 4x +5$ must be positive. It could be negative or even non-real. But it can't be $0$.

so your only restriction is that $x \ne 2\pm i$.

Also there is no order in complex numbers. $x < 2-i$ does not make sense.

So if you are working with Complex numbers the domain is all $x$ so that $x^2 -4x +5 \ne 0$ or all complex except $2\pm i$. $2$ and $3$ are within that domain so you are safe.

If you are working with Real numbers the domain is all $x$ where $x^2 -4x + 5 = (x-2)^2 + 1 > 0$. This is all real numbers. $2$ and $3$ are within that domain so you are safe.

Tags:

Inequality

Complex Numbers

Logarithms

Quadratics

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