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?
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$\begingroup$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.
$\endgroup$ 1 $\begingroup$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.
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