[Math] 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.

Answer 1

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.