Conditions for Floor of Log base b of x to equal Floor of Log base b of Floor of x/Proof 2

Proof
We have that:
 * Logarithm is Strictly Increasing

and:
 * Real Natural Logarithm Function is Continuous

Suppose that $\log_b x \in \Z$: let $\log_b x = n$, say.

Then:
 * $x = b^n$

It follows that:
 * $x \in \Z \iff b \in \Z$

Thus by McEliece's Theorem (Integer Functions):
 * $\forall x \in \R_{\ge 1}: \floor {\log_b x} = \floor {\log_b \floor x} \iff b \in \Z_{>1}$