First, we'll use the properties of logarithms to simplify the equation.

log0.5(4x-1) - log0.5(7x-3) = 0

We can combine the two logarithms using the division property of logarithms:

log0.5[(4x-1) / (7x-3)] = 0

Now, we'll convert the logarithmic equation into an exponential equation:

0.5^0 = (4x-1) / (7x-3)

Since any number raised to the power of 0 is equal to 1, we have:

1 = (4x-1) / (7x-3)

Now, let's simplify the equation further by cross multiplying:

7x - 3 = 4x - 1

Subtracting 4x from both sides, we get:

3x - 3 = -1

Adding 3 to both sides, we get:

3x = 2

Dividing by 3, we find:

x = 2/3

Therefore, the solution to the equation log0.5(4x-1) - log0.5(7x-3) = 0 is x = 2/3.