Prime gap around x expressed as $w(x)\log^{w(x)}x$

Question

For $ x $ a positive real number greater than $3 $ and not equal to a prime, there exists a unique pair of consecutive primes $ (p,q) $ such that $ p<x<q $. Let $ g(x) : =q-p $. Expressing $ g(x) $ as $ w(x)\log^{w(x)}x $ with $ w(x) $ positive, what is the best upper bound for $ w(x) $ one can get in term of $ x $?

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