log(b^x = 1/2 log b^4 + 2/3 logb27 - logb^6)

I assume you mean log(b^x) = (1/2)log(b⁴) +(2/3)log(b²⁷) - log(b⁶) and you want to find x.

Using the properties of logs (see the Formulary) we can write the right-hand side as log(b^4*(1/2)) + log(b^27*(2/3)) - log(b⁶), which is log(b²)+log(b¹⁸)-log(b⁶), which is log(b^2+18-6), which is log(b¹⁴).  Comparing this with log(b^x) we can see that x is 14.