It's not sloppiness, it's economy.

You can convert between log bases by multiplying by a constant factor. But any real-world program will also have a constant factor associated with it, depending on the specific work it is doing and the hardware it is running on. So it is usually pointless to consider constant factors when theorising about computer programs in general.

It doesn't matter in O() notation.

One answer, from a certain perspective, is that it's relative to your encoding base. It's logarithmic in operations, with the base depending on the encoding.

Another reason is because base e notation is ln, not log

This is very common. Log without further specification can be assumed to be the natural log (log e).