Conditions for Floor of Log base b of x to equal Floor of Log base b of Floor of x

Theorem
Let $b \in \R$ be a real number.


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

where $\left\lfloor{x}\right\rfloor$ denotes the floor of $x$.