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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.

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The formula you quote is from Number Theory. So for sure base $e$. – André Nicolas Feb 3 '13 at 17:37
up vote 33 down vote accepted

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.

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Some computer-science types use $\mathop{lg}$ for $\log_2$ – vonbrand Feb 3 '13 at 19:13
I've often seen lb for log2 (i.e. log binary) – wim Feb 3 '13 at 23:15
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 4 '13 at 0:21
The second term in the original expression has a log divided by the square of a log, so the base is relevant. – DJohnM Feb 4 '13 at 2:06
@wim Yes. That's the ISO 31-11 standard (lb for base 2, ln for $e$, lg for 10). – Mechanical snail Feb 4 '13 at 8:21

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 $e$

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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 3 '13 at 19:17
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.) – Jefromi Feb 3 '13 at 20:45

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.)

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