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I've always thought that $\log(x)$ is a shorthand representation of $\log_{10}(x)$. However, recently, I've noticed that some places such as Wolfram Alpha define $\log(x)$ as $\log_e(x)$, which I would write as $\ln(x)$.

So what's the standard, and what should I be using?

Beta Decay
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    Always was $\ln{x}=\log_{e}x$ and $\log{x}=\log_{10}x$, but in the last time there are professors, which write $\log{x}$ like $\ln{x}$. It's very bad, I think. – Michael Rozenberg Aug 23 '17 at 18:55
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    Since the era of slide rules has passed, base 10 logs have little practical value. However natural logs are still quite relevant for calculus. Frequently, it doesn't actually matter. You get something like $r^x = b \implies x = \frac {\log b}{\log r} $ and the result is true whichever base you choose. I preffer $\ln$ for base $e$ and to be explicit with my base if the base of the log is not $e$ and it turns out to matter. – Doug M Aug 23 '17 at 18:58
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    It highly depends on the context. Mathematicians often assume $\log$ is $\log_e$, since it is the "natural log," while $\log_{10}$ is kind of an arbitrary choice (used mostly to make base $10$ calculations easier.) – Thomas Andrews Aug 23 '17 at 18:59
  • I learned $\log$ to be base ten in school. In computer science it's base two. It differs. – Arthur Aug 23 '17 at 19:06
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    The inverse function of the solution of the differential equation $f'(x)=f(x), f(0)=1$ does not know how many fingers we have. So $\log x$ usually means $\ln x=\log_e(x)$. – Jack D'Aurizio Aug 23 '17 at 23:40

3 Answers3

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Use whatever is clear in the context:

  • No one really uses $\log_e(x)$.

  • $\ln (x)$ and $\log_{10}(x)$ are always clear. Use them if you need.

  • The only ambiguous case is $\log(x)$.

  • In advanced mathematics, $\log(x)$ always means $\log_e(x)$.

lhf
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To the extent that there's "a standard", it's probably ISO 80000-2, which prescribes the following:

  • $\log_a x$ is the logarithm to base $a$.
  • $\ln x = \log_e x$ is the logarithm to base $e$.
  • $\text{lg }x = \log_{10} x$ is the common or decimal logarithm.
  • $\text{lb }x = \log_2 x$ is the binary logarithm.
  • $\log x$ is "used when the base does not need to be specified" (e.g., in a proportionality), and "shall not be used in place of $\ln x$, $\text{lg }x$, $\text{lb }x$, or $\log_e x$, $\log_{10} x$, $\log_2 x$."

That said, remarkably few scientists, engineers, and academics are all that concerned with ISO standards, and they generally flout these rules with wanton and reckless disregard for the dire consequences that may arise from violating these hallowed and revered standards. In other words, there really isn't a de facto standard (even if there is a de jure one), and one should always keep this ambiguity in mind and attempt to glean the meaning from context when one encounters $\log x$ in a text.

Michael Seifert
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It depends.

As a math person we exclusively write $\ln x$. In the statistics books I studied out of for regression and time series, we used $\log x$ to mean the natural log.

Of course, in high school courses, $\log x$ means $\log_{10} x$. The convention differs and the author of whatever work you're looking at should specify this convention.

(Personally I write $\log$ instead of $\ln$ and I always tell my students this.)

Sean Roberson
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