how do you get the (x^7/2) from

Question

1/2 loga X + 3 loga Y - 4 loga X

Answer 1

assuming all logs are base a, and fixing your typo,
1/2 logx + 3logx - 4logx
= log x^(1/2) + log x^3 - log x^4
= log (x^(1/2) * x^3 / x^4)
= log x^(1/2 + 3 - 4)
= log x^(-1/2)
Clearly that is not all that's wrong.

Go look at the problem again, and recall that
log x^n = n logx
log(x*y) = logx + logy