I don't really understand what you're asking for. You can easily just verify the claim for yourself.

The limit as x goes to infinity of (ln x) / (x^k) is zero for all positive k. So if you want to approximate the behavior of a logarithm with a function of the form f(x) = x^k, k must be greater than 0 [because x^0 = o(ln x)], but less than every positive real number. This is the definition of an infinitesimal.

That's why it makes sense to say that x (log x)^3 is equivalent to x^{1.000000...[infinite 0s]...1}. Though in this analogy, it's more like x^{1.000000...3}.