(Cross-posted from cstheory-stackexchange)

The following fact seems to be used implicitly in cs theory, particularly algorithms. Given a RAM machine $M$ running in time $O(f(n))$, another RAM machine $M′$ can simulate $M$ in time $O(f(n))$. This differs from the case for Turing machines, where $M′$ may require $O(f(n) \log(f(n))$ time.

I say this is often used implicitly because many papers will simply say something like "run $M$, but keep track of certain auxiliary information as you do so". This is really simulating $M$, but for RAM machines the distinction is not so important because running times are not (asymptotically) affected.

Is there a reference for this theorem? I am summarizing the situation correctly?