No. Usually log without further specification is base10.

Except in mathematics and physics, where it usually is base e.

Except sometimes in computer science, where it can be base 2.

But there are more of those weirdnesses: "ld" can be "log decimal", so base 10; or "logarithmus dualis", so base 2. Base 2 is also sometimes written as "lb" (log binary). You really need to know the conventions of the field/subfield to know which is which.

In this case, and in many other cases, log without further specification is meant to be understood as just "it is logarithmic" without further specification. With big O, we don't differentiate between log bases, just as we don't differentiate between scale factors. In fact, converting from one log base to another is just multiplying by a constant.

In information theory it usually means log base 2

no, that's what ln is for