  In this paper, we show that for any integer $a \geq 2$, each of the intervals
$[a^k , a^{k + 1})$ ($k \in \mathbb{N}$) contains either $\left\lfloor
\frac{\log a}{\log\Phi}\right\rfloor$ or $\left\lceil \frac{\log
a}{\log\Phi}\right\rceil$ Fibonacci numbers. In addition, the density (in
$\mathbb{N}$) of the set of the all natural numbers $k$ for which the interval
$[a^k , a^{k + 1})$ contains exactly $\left\lfloor \frac{\log
a}{\log\Phi}\right\rfloor$ Fibonacci numbers is equal to $\left(1 -
\left\langle \frac{\log a}{\log\Phi}\right\rangle\right)$ and the density of
the set of the all natural numbers $k$ for which the interval $[a^k , a^{k +
1})$ contains exactly $\left\lceil \frac{\log a}{\log\Phi}\right\rceil$
Fibonacci numbers is equal to $\left\langle \frac{\log
a}{\log\Phi}\right\rangle$.
