This is admittedly, probably an easy question for the right person here, but I cannot seem to track down an answer. The question itself may not be hard, but the answer is crucial to a math paper I am writing (and I don't know enough number theory).

True or false: There are at least exp$(r)$ irreducible polynomials in $\mathbb{F}_2[x]$ of degree $r$ or less. [I don't care about the hidden constants in the exp$(r)$-notation; could be $2^{\frac{r}{100}}$ or it could be $\frac{2^r}{r}$.]

If True a reference would be terrific.

I am aware that there are roughly $n/\log n$ primes of size $n$ or less for $n$ large, I am trying to adapt that to a polynomial ring where the coefficients are in a characteristic 2 field.