What is the asymptotic bound of the floor of the logarithmic function raised to the power of the base?

Question

In other words, what is the big-O, or big-$\Theta$ of the function $ a^{\lfloor{\log_an} \rfloor} $?

I suspect its asymptotic upper bound is $a^n$, but I cannot prove it.

Answer 1

Expanding Wouter's comment, for $a \gt 1$

$$\frac{n}{a} = a^{\log_a (n) -1 } \leq a^{\lfloor \log_a (n) \rfloor}\leq a^{\log_a (n)}=n$$

So $a^{\lfloor \log_a (n) \rfloor}$ is $\Theta(n)$ for $a > 1$

A similar argument (reversing the inequalities) would show this is also the case for $0 \lt a \lt 1$