Log base 2 is just how many times we can you recursively split something in half, which is not that complicated of a concept. However, his log example is base 10 and if he was trying to figure out these things as base 10 it would be super confusing and not make any sense.
log (base X) Y === log (base e) Y / log (base e) X
So the difference between log (base 2) and log (base 10) is just the difference between being divided by log (base e) 2 or log (base e) 10. Since those are both constant factors, big-O notation doesn’t care.