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I am currently taking a course in differential equations and got back the results of my first exam. I was surprised to see that I had points taken off for writing the natural logarithm of $x$ as $\log(x)$ instead of $\ln(x)$.

Is the notation $\log(x)$ for the natural log of $x$ incorrect? I was under the impression that it is not uncommon for mathematicians to write the natural logarithm in this way.

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    $\begingroup$ $\log(x)$ denotes $\log_{10}(x)$ to some, and $\log_e(x)=\ln(x)$ to most. $\endgroup$ – Dave Feb 26 '18 at 19:19
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    $\begingroup$ I agree that it is not incommon. I would even say that in pure mathematics $\log$ is more common than $\ln$ for the natural logarithm (this is just my opinion). But maybe you used the other notation in your course? $\log$ is sometimes used as the $10-$logarithm. $\endgroup$ – 57Jimmy Feb 26 '18 at 19:21
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    $\begingroup$ See also math.stackexchange.com/q/2403836/589 $\endgroup$ – lhf Feb 26 '18 at 19:21
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    $\begingroup$ In pure mathematics, $\log$ unambiguously means $\log_e$. In engineering, $\log$ means $\log_{10}$. I think it was stupid for your instructor to deduct points for something that is correct. $\endgroup$ – A. Goodier Feb 26 '18 at 19:22
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    $\begingroup$ Though log$(x)$ is often taken to denote log$_10(x)$ it's safer to make the base explicit. I've always taken ln$(x)$ to mean log$_e(x)$. $\endgroup$ – jim Feb 26 '18 at 19:37
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It's normally context-dependent. To a mathematician, $\log(x)$ means $\log_{10}(x)$, and the natural log is always $\ln(x)$. To a physicist, $\log(x)$ is the natural logarithm. To an engineer (not computer), $\log(x)=\log_{10}(x)$. To a computer engineer, $\log(x)=\log_2(x)$.

Bottom line: if you want to be completely unambiguous, and you're not sure of the context, write $\log_b(x)$, where $b=e, 2,$ or $10$. On the other hand, $\ln(x)$ always means $\log_e(x)$, so that's also unambiguous.

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    $\begingroup$ I would be careful stereotyping here. (Many mathematicians write $\log$ for the natural logarithm, if they're in a setting where $\log_{10}$ is never useful. The convention $\log = \log_2$ is also adopted by many computer scientists, including ones working in theory that are certainly mathematicians. And regardless of the field, people who do lots of calculations in software will adopt the conventions of that software.) I agree a hundred percent with your bottom line, though. $\endgroup$ – Misha Lavrov Feb 26 '18 at 19:27
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    $\begingroup$ The natural logaritm is always $\log x$ to many pure mathematicians too! (As illustrated by the joke about the drowning analytic number theorist who said “log-log-log-log”...) $\endgroup$ – Hans Lundmark Feb 26 '18 at 19:28
  • $\begingroup$ @Misha: Sure, there are exceptions to stereotypes, but I think these are valid in my general experience. And point taken about software, as well. As Dr. Sonnhard Graubner pointed out in another answer, the Wolfram Language uses $\text{Log}[x]$ for $\ln(x)$. As Stephen Wolfram is a physicist, I'm not totally surprised. $\endgroup$ – Adrian Keister Feb 26 '18 at 19:33
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    $\begingroup$ Surely to a mathematician, only the natural logarithm exists, and the idea of using logs to other basis is completely incomprehensible. $\endgroup$ – Lord Shark the Unknown Feb 26 '18 at 19:33
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    $\begingroup$ @LordSharktheUnknown But to a discrete mathematician, $e=2$. $\endgroup$ – Misha Lavrov Feb 26 '18 at 19:34
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You are right, but the alternative notation, at least as frequent, is to use $\ln$ for natural log, $\log$ for base $10$ and $\lg$ for base $2$.

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when you are looking in Wolfram Alpha then $\log(x)$ denotes the natural logarithm and $\lg(x)$ the logarithm to the base $10$

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