Why does Wolfram Alpha handle $\log$ and $\ln$ the same?

Question

I thought $\log(n)$ was like $10^x = n$ and $\ln(n)$ was $e^x = n$. But when I do $\ln(80)$, it gives me the answer for $\log$. Why is that?

Answer 1

It's simply a matter of definitions.

In all fields, $\ln$ means the natural log, or log base $e$, so that $\ln n = x$ whenever $e^x = n$. In engineering (and high school), $\log$ usually means the common log, or log base $10$, so that $\log n = x$ whenever $10^x = n$.

However, it happens that in higher mathematics, the common log just isn't very important. So for convenience, mathematicians often use the notation $\log$ to represent the natural log. Wolfram Alpha does things the same way.

Answer 2

In theoretical mathematics and in many programming languages "$\log$" usually means natural (base-$e$) logarithm. That's the only logarithm that's important for most theoretical purposes.

Sometimes I've written "log" in an expression on a blackboard and a student has asked "Do you mean logarithm, or natural logarithm?", and I've said "Yes". I dislike "ln" because I suspect it of causing some people to think a natural logarithm is something different from a logarithm, as evidenced by questions like that one.