It is easy to create a sequence $\{m_n\}$ for which the order of $2\pmod{m_n}$ is as small as possible, i.e. it is about $\log_2(m_n)$. For example $m_n=2^n-1$ is an appropriate sequence. But if I consider the bases 2 and 3, then it seems rather complicated to create a sequence $\{m_n\}$, for which the orders of both 2 and 3 are "about" $\log(m_n)$.
Great amount of examples show that for any $1<c\in\mathbb{R}$, there are only finitely many integers $m$, such that the order of $2$ and $3$ are less than $c\cdot\log(m)$.
However, it seems that $(\log(m))^2$ is a reasonable choice for the upper bound of the orders. Considering the examples, it seems that there are infinitely many integers $m$ for which the orders of $2$ and $3$ are less than $\log(m)^2$. I would really love this to be true.
The "next" step in the exponent is generally true, a theorem of Erdős, Pomerance and Schmutz about the small values of the Carmichael function states that there are infinitely many integers $m$, such that the order of any element in the multiplicative group of the ring $\mathbb{Z}/m\mathbb{Z}$ is of order less than $$\log(m)^{c\log\log\log(m)}.$$
I did not find any papers or results dealing with such a problem. Not necessarily with only the bases 2 and 3. I am interested in any results about fixed number of bases with small orders. I am sure that it is already investigated by many mathematicians.
Could anyone help with this or suggest some papers?