We can use the following logarithmic properties to rewrite the expression as a single logarithm:
log a + log b = log(ab)
log a - log b = log(a/b)
n log a = log (a^n)
Using these properties, we can rewrite the expression as:
1/2logx - 3log(sin2x) + 2
= log(x^(1/2)) - log((sin2x)^3) + log(e^2)
= log[(x^(1/2))(e^2)/((sin2x)^3)]
Therefore, the expression as a single logarithm is:
log[(x^(1/2))(e^2)/((sin2x)^3)]