3. Solve in $\mathbb{R}$ the equation: $\log _{a}[x]=\left[\log _{a} x\right], a \in \mathbb{N}, a \geq 2$.
3. Solve the equation in $\mathbb{R}$: $\log _{a}[x]=\left[\log _{a} x\right], a \in \mathbb{N}, a \geq 2$.

Barem: Conditions for existence: $\quad x \in[1,+\infty)$

$$
a^{k}+1 \leq a^{k+1} \text { for } a \geq 2 \text { and } k \in \mathbb{N} \Rightarrow x \in \bigcup_{k \in \mathbb{N}}\left[a^{k}, a^{k}+1\right)
$$