this means that 2x-3 must be greater than 0.
in the expression 2x-3, solve for x to get x > 3/2.
the function will give you a real value of y only if the value of x is greater than 3/2.
your domain is all real values of x such that x > 3/2.
by definition of logs, y = logb(x) if and only if b^y = x.
your equation is y = log2(2x-3).
this is true if and only if 2^y = 2x-3.
since 2x-3 must be greater than 0, then 2^y = 2x-3 is equivalent to 2^y > 0.
2^y will always be greater than 0 for all real values of y.
since 2^y will always be greater than 0, regardless if y is itself > 0 or = 0 or < 0, then your range is all real values of y.
if you graph the equation of y = log2(3x-2), it will look like this:
as the value of x approaches 1.5 from the right on the graph, the value of y becomes more and more negative.
your domain is all values of x such that x >= 3/2.
your range is all real values of y.