That isn't just what's meant, that's also what's written.

The way that it was written didn't disambiguate whether or not the log is part of the exponent. Putting the * in there disambiguated that.

It doesn't. The multiplication is implied and it would have the same precedence as explicitly putting * there. So either both are ambiguous or neither is (and it isn't, the poster was just mistaken).

I'm not sure why you think that it is implied.

I know that I have a lot of math background, and my brain read that log(n) as part of the exponent. I figure that if I read it that way, you have to expect that others will as well.

No matter how you read a^b log(c), the multiplication is implied. The original confusion is whether it's to be read as (a^b)log(c) or a^(b log(c)). a^b*log(c) simply removes the implied multiplication, but it does not change whether that should be read as (a^b)*log(c) or a^(b*log(c)) and thus does not resolve the original confusion.