To solve the equation 2log₀,₅x = log₀,₅(2x-x), we first need to use the properties of logarithms to simplify.

Using the power rule of logarithms, we can write 2log₀,₅x as log₀,₅(x^2). Similarly, we can write log₀,₅(2x-x) as log₀,₅(x).

So the equation becomes:
log₀,₅(x^2) = log₀,₅(x)

Now, we can drop the logarithms from both sides and set the expressions inside them equal to each other:
x^2 = x

This is a quadratic equation, so we need to rearrange it into standard form and solve:
x^2 - x = 0
x(x - 1) = 0

Setting each factor to zero gives us two possible solutions:
x = 0 or x = 1

Therefore, the solutions to the equation 2log₀,₅x = log₀,₅(2x-x) are x = 0 and x = 1.