I saw a lot of answers on this site, people use $\log(f(x))$ to represent logarithm in base $e$. I have read some questions and answers about it like these:
Which is more preferable to write $\log(x)$ or $\ln(x)$
Should I assume $\log(x)$ to be $\log_e(x)$ or $\log_{10}(x)$?
From above posts, I realized in pure mathematics for advanced level it is common to use $\log(x)$ to denote logarithm in base $e$.
My question is: why we are allowed to do this? if $\log(x)$ use to denote logarithm in base $10$ too, then why we use this instead of $\ln(x)$? As far as I know the purpose of mathematic is explaining something in most clear way and unambiguously so we should use notation proper for this purpose. but suppose the subject we are talking about has nothing to do with logarithm in base $10$ and we use $\log(x)$ to denote $\ln(x)$. but still $\log(x)$ has two meaning (logarithm in base $10$ or in base $e$)therefor it has contrast with the purpose I mentioned. why don't we avoid using this notation?
Best Answer
We tend to introduce $\log(x) = \log_{10}(x)$ because before you know anything about logarithms it's most convenient to think about things in terms of base $10$. Once you learn enough about logs, you know that base $e$ is really the most convenient to work with, but at that point you've probably been introduced to the notation $\ln(x) = \log_{e}(x)$. For many mathematicians, there comes a point when you realize the only logarithm you really care about is base $e$, so whenever possible it's convenient to just redefine $\log(x) = \log_{e}(x)$.