Should I assume $\log(x)$ to be $\log_e(x)$ or $\log_{10}(x)$?

Use whatever is clear in the context:

• No one really uses $\log_e(x)$.

• $\ln (x)$ and $\log_{10}(x)$ are always clear. Use them if you need.

• The only ambiguous case is $\log(x)$.

• In advanced mathematics, $\log(x)$ always means $\log_e(x)$.

To the extent that there's "a standard", it's probably ISO 80000-2, which prescribes the following:

• $\log_a x$ is the logarithm to base $a$.
• $\ln x = \log_e x$ is the logarithm to base $e$.
• $\text{lg }x = \log_{10} x$ is the common or decimal logarithm.
• $\text{lb }x = \log_2 x$ is the binary logarithm.
• $\log x$ is "used when the base does not need to be specified" (e.g., in a proportionality), and "shall not be used in place of $\ln x$, $\text{lg }x$, $\text{lb }x$, or $\log_e x$, $\log_{10} x$, $\log_2 x$."

That said, remarkably few scientists, engineers, and academics are all that concerned with ISO standards, and they generally flout these rules with wanton and reckless disregard for the dire consequences that may arise from violating these hallowed and revered standards. In other words, there really isn't a de facto standard (even if there is a de jure one), and one should always keep this ambiguity in mind and attempt to glean the meaning from context when one encounters $\log x$ in a text.