a) To solve the equation log₀,₅(x²-3x) = -2, we can rewrite it in exponential form: 0,5^(-2) = x²-3x. Simplify: 4 = x²-3x. Rearrange: x²-3x-4 = 0. This is a quadratic equation, which can be factored as (x-4)(x+1) = 0. Therefore, the solutions are x=4 and x=-1.

b) For the equation log₂²(x-2) - log₂(x-2) = 2, we can combine the logs using the identity loga(b) - loga(c) = loga(b/c). Log₂²(x-2)/(x-2) = 2. That simplifies to log₂(x-2) = 2. Rewrite in exponential form: 2^2 = x-2. Simplify: 4 = x-2. Solve for x: x = 6.

c) To solve the inequality log₃(x²+2x) < 1, we can rewrite it in exponential form: 3^1 > x²+2x. Simplify: 3 > x²+2x. Rearrange: x²+2x-3 < 0. We can factor this quadratic to get (x+3)(x-1) < 0. The solutions are x < -3 or 1 < x < 0.

d) The inequality log₁/₃(0,1x-5,2) > 2 can be rewritten in exponential form: 1/3^2 > 0,1x-5,2. Simplify: 1/9 > 0,1x-5,2. Rearrange: 0,1x-5,2 < 1/9. This simplifies to 0,1x-5,2 < 1/9. Solving for x gives x > 59/10.