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

Solution 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.

Solution 2:

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 $e$

Solution 3:

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.)