jmc

We can have ${\log }_{7}343=3$ and ${\log }_{7}2401=4$. Since ${\log }_{7}x$ increases as $x$ increases, we know that ${\log }_{7}343<{\log }_{7}2400<{\log }_{7}2401$, meaning $3<{\log }_{7}2400<4$. Moreover, we can see that $2400$ is much closer to $2401$ than to $343,$ so it stands to reason that ${\log }_{7}2400$ rounded to the nearest integer is $\boxed{4}.$