Conditions for Floor of Log base b of x to equal Floor of Log base b of Floor of x/Proof 1

Necessary Condition
Let:
 * $\forall x \in \R_{\ge 1}: \floor {\log_b x} = \floor {\log_b \floor x}$

Let $x = b$.

Then:
 * $\floor {\log_b b} = \floor {\log_b \floor b}$

$b \notin \Z$.

Thus by Proof by Contradiction:
 * $b \in \Z$

But for $\log_b$ to be defined, $b > 0$ and $b \ne 1$.

Hence:
 * $b \in \Z_{> 1}$

Sufficient Condition
Let $b \in \Z_{> 1}$.

Let $\floor {\log_b x} = n$.

Then: