To solve lg((x-5)^2) = 22, we first need to rewrite the equation in exponential form. The base of the logarithm lg is 10, so the equation becomes:

10^22 = (x-5)^2

Now we can solve for x:

10^22 = x^2 - 10x + 25
x^2 - 10x + 25 - 10^22 = 0

This is a quadratic equation that can be solved using the quadratic formula:

x = [10 ± √(10^2 - 41(25-10^22))] / 2
x = [10 ± √(100 - 100 + 410^22)] / 2
x = [10 ± √(410^22)] / 2
x = [10 ± 2√(10^22)] / 2
x = 5 ± √(10^22)

So the two possible solutions for x are: x = 5 + √(10^22) and x = 5 - √(10^22).

To solve x^-4 + log5(5x) = 625, we first simplify the equation by combining the logarithmic term with the exponent term:

5^(x^-4) * 5^(log5(5x)) = 5^625

5^(x^-4 + log5(5x)) = 5^625

Now we can rewrite the equation using the properties of logarithms:

5^[(1/x^4) + 1] = 5^625

Since the base is the same, we can set the exponents equal to each other:

(1/x^4) + 1 = 625

Now we can solve for x:

1/x^4 + 1 = 625
1/x^4 = 624
x^4 = 1/624
x = (1/624)^(1/4)

Therefore, the solution for x is x = (1/624)^(1/4).