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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?

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    $\begingroup$ It really doesn't matter for most purposes what base you use since one is just a scalar multiple of the other. Using base e is just most convenient in a lot of cases $\endgroup$
    – Ben
    Nov 23, 2020 at 1:59
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    $\begingroup$ To my understanding, the base of $\log(x)$ is just whatever is most common in your field or relevant to it. In mathematics, that'd be $e$ (for which $\ln(x)$ exists); in computer science, that would be $2$ (for which $\operatorname{lb}(x)$ exists); in other places, it would be base $10$. It's one of those issues where you have to be on the same page as everyone else, sadly, so IMO it's generally better to be explicit with what base you're using rather than just using $\log(x)$. Though, at least as Anon pointed out, the change of base formula admits them all being scalar multiples of each other $\endgroup$ Nov 23, 2020 at 2:01
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    $\begingroup$ I would say it's much for the same reasons that mathematicians cannot agree whether $0$ is a natural number or not. Unfortunately, I think mathematicians also disagree upon why they can't agree on this matter. $\endgroup$ Nov 23, 2020 at 2:02
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    $\begingroup$ "What sound does a drowning analytic number theorist make?" Answer: "log log log." Let's face it - "ln ln ln" wouldn't work. $\endgroup$
    – peter a g
    Nov 23, 2020 at 2:03
  • $\begingroup$ @EeveeTrainer: I’ve never seen $\operatorname{lb}x$; I consider $\operatorname{lg}x$ the standard alternative to $\log_2x$. $\endgroup$ Nov 23, 2020 at 2:16

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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 $\log(x) = \log_{e}(x)$.

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In pure mathematics, we hardly ever need to use $\log_{10}$.

It's true that using $\ln$ would probably be more clear, but we tend to read out "$\ln$" as "log" anyway (out loud), so there is an argument to be made to just use $\log$ since it's the most natural notation which reflects how we communicate.

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  • $\begingroup$ I have always read $\ln$ aloud as lin, a practice that I picked up from others decades ago. $\endgroup$ Nov 23, 2020 at 2:15
  • $\begingroup$ I have always read it as el-en. Maybe there are people who speak it as log. $\endgroup$ Nov 23, 2020 at 3:21
  • $\begingroup$ @ultralegend5385 me too $\endgroup$
    – Etemon
    Nov 23, 2020 at 3:28
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It really depends. Mostly in physics $\log(x)$ is assumed as base e but to be safe you should probably us $\ln(x)$. It highly depends on the context. In elementary math, $\log(x)$ is assumed as base $10$.

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