An interesting Bayesian question

Suppose a random sample of size n is from uniform distribution U(0, exp(gamma)), suppose the prior distribution of gamma is N(a, b), how to find the posterior density function of gamma given (y1, ..., yn)?

I followed the routine of writing the posterior density function as proportional to the product of likelihood, which is (exp(gamma))^(-n), and the prior, which is N(a, b). But later I found that the domain of gamma should be greater than log(max(y1, ..., yn)) which makes the problem become difficult. Anyone can help? Thanks.