When log is written without a base, is the equation normally referring to log base 10 or natural log?

Question

For example, this question presents the equation

$$\omega(n) < \frac{\log n}{\log \log n} + 1.4573 \frac{\log n}{(\log \log n)^{2}},$$

but I'm not entirely sure if this is referring to log base $10$ or the natural logarithm.

The formula you quote is from Number Theory. So for sure base $e$. — André Nicolas, Feb 03 '13 at 17:37

score 92 · Accepted Answer · edited Jun 12 '20 at 10:38

92

In mathematics, $\log n$ is most often taken to be the natural logarithm. The notation $\ln(x)$ not seen frequently past multivariable calculus, since the logarithm base $10$ finds relatively little use.

This Wikipedia page gives a classification of where each definition, that is base $2$, $e$ and $10$, are used:

$\log (x)$ refers to $\log_2 (x)$ in computer science and information theory.

$\log(x)$ refers to $\log_e(x)$ or the natural logrithm in mathematical analysis, physics, chemistry, statistics, economics, and some engineering fields.

$\log(x)$ refers to $\log_{10}(x)$ in various engineering fields, logarithm tables, and handheld calculators.

edited Jun 12 '20 at 10:38

Community

1

answered Feb 03 '13 at 17:33

Eric Naslund

72,099

9

Some computer-science types use $\mathop{lg}$ for $\log_2$ – vonbrand Feb 03 '13 at 19:13
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I've often seen lb for log2 (i.e. log binary) – wim Feb 03 '13 at 23:15
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To add on this, in terms of computational complexity all the bases are the same as they differ by a constant. So it's actually irrelevant which base you chose for your logarithm in the aforementioned question (about the prime numbers). – Asaf Karagila Feb 04 '13 at 00:21
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The second term in the original expression has a log divided by the square of a log, so the base is relevant. – DJohnM Feb 04 '13 at 02:06
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@wim Yes. That's the ISO 31-11 standard (lb for base 2, ln for $e$, lg for 10). – Mechanical snail Feb 04 '13 at 08:21
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Given this many possible contexts, it seems unsuitable for the base to be implicit. – ardave Dec 29 '16 at 19:11
My economics microeconomics teacher used $\log x$ to denote $\log_{10} x$ . I have no idea why and he got all defensive when I asked. – user1561358 Nov 29 '17 at 09:50
Note that log 10 is use extensively in chemistry for pH – Juan Perez May 09 '22 at 14:38

score 19 · Answer 2 · answered Feb 03 '13 at 17:34

19

Depending on the subject, it can be base $10$, base $e$, or base $2$. Base $2$ is common in computer science. Base $10$ is popular in engineering (think decibels). I would take this to be base $e$

answered Feb 03 '13 at 17:34

Ross Millikan

374,822

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Oh, this new generation. For us old-timers, who grew up with slide rules and logarithm tables, $\log$ was $\log_{10}$ by default. $\ln$ was used only in the weird formulas used by mathematicians, to be translated into $\log_{10}$ as soon as possible [Chemical engineer by training, in the '70s]. – vonbrand Feb 03 '13 at 19:17
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Base 10 also shows up reasonably often in the less mathy sciences. (vonbrand's comment generally describes the kind of fields where it might be used - if they think math is weird, there's a good chance they use base 10.) – Cascabel Feb 03 '13 at 20:45

score 13 · Answer 3 · answered Feb 04 '13 at 08:32

In some cases, "$\log$" can refer to a logarithm with an indefinite base.

Suppose we're taking the logarithm (base $b$, where $b>0$ is constant) of some variable. Recall the identity

$$\log_b(x) = \frac{\log_c(x)}{\log_c(b)}$$

The base-$b$ logarithm can be expressed as a constant factor times the logarithm to any other base $c>0$. In some domains, particularly asymptotic analysis, we don't care about constant factors—which means that it doesn't matter what base we pick. So we can unambiguously write $Θ(\log(n))$ without specifying the base.

(This does not apply to the specific usage in the question, which is about an upper bound for all $n$. Obviously constant factors matter there.)

score 0 · Answer 4 · answered Apr 13 '21 at 19:31

0

If you come from a computer science background, then:

log(x) = log$_{10}$(x)
ln(x) = log$_{e}$(x)
lg(x) = log$_{2}$(x)

If a computer scientist is using log(x) with a base that is not 10, then he/she will usually point it out clearly.

answered Apr 13 '21 at 19:31

stackoverflowuser2010

1,023

score 0 · Answer 5 · answered Apr 15 '21 at 05:13

Generally it is bad practice to write logarithms without the base especially if the context hasn't been sufficiently set up. If a base is to be implied, then it is better to use one of the variants, such as $ln()$ for base $e$, $lg()$ for base $2$, etc.

But there is literature out there that violates the above rule, so yes, you'll need to infer the base from the context.

Having said that, there are cases when a missing base is intentional and acceptable, implying that it is applicable regardless of what the base is. For instance, when I say:

$$log_{2}(10) = log(10)/log(2)$$

and you notice that the base is omitted in the RHS, it means that it is applicable to any base in the RHS (of course, as long as the same base is used in both numerator and denominator).

To me, that is the only acceptable case when one can omit the base.

score 0 · Answer 6 · answered Dec 28 '21 at 21:17

0

In many programming languages, log is the natural logarithm. There are often variants for log2 and log10.

Checked in C, C++, Java, JavaScript, R, Python

answered Dec 28 '21 at 21:17

David Gilbertson

101

When log is written without a base, is the equation normally referring to log base 10 or natural log?

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