To solve the given equation, we can apply the properties of logarithms. Let's solve step by step:

Log 1/2 (3x - 5) = log 1/2 (x^2 - 3)

Since the bases of the logarithms are the same, we can set the arguments equal to each other:

3x - 5 = x^2 - 3

Rearranging the equation gives us a quadratic equation:

x^2 - 3x - 2 = 0

Log 2 (x^2 - 3x) = 2

Applying the power rule of logarithms, we get:

x^2 - 3x = 2^2

x^2 - 3x = 4

Log 2x + log 2 (x-3) = 2

Using the product rule of logarithms, we simplify to:

log 2x(x-3) = 2

Taking antilogarithm on both sides:

2x(x-3) = 2^2

2x^2 - 6x = 4

lg(x^2) - 2lg(x) - 3 = 0

This equation can be rewritten as:

log(x^2) - 2log(x) - 3 = 0

2log(x) - log(x^2) = 3

log(x^2) - log(x^2) = 3

0 = 3

Therefore, the given equation does not have a solution as the last step leads to an erroneous statement.