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

Theorem

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

$\forall x \in \R_{\ge 1}: \floor {\log_b x} = \floor {\log_b \floor x} \iff b \in \Z_{> 1}$

where $\floor x$ denotes the floor of $x$.

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):

$\forall x \in \R_{\ge 1}: \floor {\log_b x} = \floor {\log_b \floor x} \iff b \in \Z_{>1}$

$\blacksquare$

Sources

• 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.4$: Integer Functions and Elementary Number Theory: Exercise $34$