Take some infinite hypernatural number, $M$, and consider the integers (finite and infinite) less than or equal to $M$. There are uncountably many. Then consider $\log_2 M$. Is there a straightforward way to understand the cardinality? Could it be countable?
If $M$ is nonstandard, so is $\log_2(M)$, so the same reasoning shows that there are uncountably many nonstandard integers $<\log_2(M)$ as well.