Lower bound for the multiplicative order of a fixed integer $a$ modulo $n$, as $n$ grows large

Question

Let $a \geq 2$ be fixed. Is there any good lower bound known for the multiplicative order $\text{ord}_n(a)$ of $a$ modulo $n$ (with $n,a$ relatively prime) as $n$ grows large? Clearly $\text{ord}_n(a)$ should be at least of the order of $\log n$. But can we do better? What about in the case when $n$ runs over primes?