# Ln(x) or log(x)?

## ln(x) or log(x)?

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• Total voters
48
Which do you write as your default notation for the logarithm to the base e, i.e. the natural logarithm? Personally, I write it as log(x), and I cringe whenever I see ln(x) used. This is mostly because just about no mathematician in the world cares about the base 10 logarithm. As far as I have learned, log(x) means base e to mathematicians, base 10 to engineers, and base 2 to computer scientists. Funny thing is, most calculators are made my engineers, which is why they have log(x) as the base 10, and ln(x).

I'm a mathematician, but I prefer Ln for base e logarithm and LOG for base 10 logarithm. The reason being, that I don't have to bother with writing the subscript to differentiate between them. Plus, it keeps me in line with the way it's done on calculators.

I just noticed ... this is a "log poll" (not to be confused with a "log pole")!

(Sorry, I couldn't resist!)

Eh I'm a math major and I've gotten used to writing log, though the convention is a bit silly. Obviously ln is unambiguous to just about everyone, whereas log is not; it's that simple. Also two caveats from experience:

1. Calculus courses do still teach the derivative of say the base 10 log, but unsurprisingly it's obtained by first converting to the natural log.

2. In complex analysis, it's important to distinguish log as a potentially multi-valued complex function (i.e. when we don't define the principal branch) versus taking the real-valued natural logarithm. Here, I always write $\log (re^{i\theta}) = \ln r + i\theta + 2\pi i k,$ because it is actually quite easy to screw this up.

nicksauce
Homework Helper
I definitely prefer "log" to refer to base e, and specify the base if I happen to be using another. It seems more elegant, I guess.

arildno
Homework Helper
Gold Member
Dearly Missed
Well, in my maths education, ln(x) was standard, so I've gotten used to to that.
I do see, however, that the most natural logarithm may have an even stronger claim on the notation log(x), or lg(x) than the logarithm that was first developed&widely used (i.e, the Briggsian, since Napier's scheme was quite different, and haven't been used after him)

It seems more elegant, I guess.

Indeed.

I prefer ln(x), simply because it takes less space and time to write. When you take up as much space as I do when writing, these things become important!

Char. Limit
Gold Member
I prefer log(x), because the natural log deserves the place of primary log! We can call the log base ten ld(x), for "logarithmus decimalis".

$$\ln(x)$$ is a convenient notation for engineers and maybe physicists, chemists, and other scientists, but i also wish it weren't there (and it wasn't there for my first calculus class).

i think it should be $$\log_{a}(x)$$ with the default for missing $a$ to be $a=e$.

there is nothing wrong with $$log_{10}(x)$$ if that is what you want. and i never use log without an explicit base, unless it's base e.

danago
Gold Member
The first time i encountered logarithms, my textbook introduced the natural log function as ln(x), so that is what i have used ever since.

I use ln(x) whenever e is the base, and lg(x) when 10 is the base. If any other number is the base, I use loga where a is the base.

It depends on whether I'm in my first analysis course or my first ODE course =P

But seriously...
I agree that ln(x) is unambiguous for the natural log.
I think that log(x) is ALWAYS ambiguous, and that if you write log(x) the context should make it clear which one you mean, or you should write the base down. My impression is that this is accepted wisdom, and that it works well.

When you write log you have to define the base, which makes it a more complicated function than the ln, which has e as a base.

btw: why would one want to use a log with 10 as a base? From what I have seen ln is almost always a good solution.

btw: why would one want to use a log with 10 as a base? From what I have seen ln is almost always a good solution.

You are right.
I suppose it's because the log function is easier to understand for those who have never worked with logarithms, if you use 10 as a base. So using 10 as a base could be a way to learn what the function means and learn about its properties. For those who know this well, it is recommended to let the base be e.

I was taught log in the ninth grade, where we used base 10. In fact, we never considered any other base, let alone one that requires Calculus. But, it was In(x) in college physics.

Worrying about the logic is not all that common in math terms. I have met engineers who thought the term "imaginary numbers," was unfortunate, since they were as real as real numbers. For centuries, "Fermat’s Last Theorem," was simply a conjecture

Historically, log X is to the base 10. That was how Briggs wrote the first log book in 1617. However, Napier in 1618 evidently recognized the natural log base. But e, Euler's number, was first written up in 1736.

Leibenz first wrote on the Calculus in 1675, before that time, I doubt there was much use to In X.

As for the logic of math terms, we have "The Method of Cardan." However, Cardan himself admitted to stealing the idea from Tartaglia. This could make some math teachers wary of the term, but as one professor put it, Cardan first published and that is where the credit belongs.

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If I had a say in that matter, I would define the logarithm as follows.

Base e: ln or log
Base 10: lg
Base 2: ld
Base a: log_a