Migration wage differential, Borjas (2013) textbook
Yes, that is an approximation one does. You have stumbled upon the log-approximation, which is very commonly used in Economics.
It states that $log(1+x)≈x$, for very small $x$. Hence, typically $x$ is a fraction. It is also used in macro to approximate growth rates.
There are lots of resources out there that explain it, e.g. here.
The way I remember it is that $log(1) =0$ so $log(1+x)≈x$ for small $x$. This is mathematically totally wrong, but I find it to be a useful memory device personally.