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:
 * 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):
 * $\forall x \in \R_{\ge 1}: \left\lfloor{\log_b x}\right\rfloor = \left\lfloor{\log_b \left\lfloor{x}\right\rfloor}\right\rfloor \iff b \in \Z_{> 1}$