# Why do people refer ln(x) as log(x) ?

Why do people refer "ln(x)" as "log(x)"?

Like wlframalpha does it and my professor too.

My professor dosen't say "the natural log of x", he just says 'log x'

Why??

micromass
Staff Emeritus
Homework Helper

Like wlframalpha does it and my professor too.

My professor dosen't say "the natural log of x", he just says 'log x'

Why??

I do it too I think it is because you never actually use the logarithm with base 10 in (pure) math. The important function in calculus is ln(x) not $\log_{10}(x)$. Therefore I find that it makes sense to say log(x) instead of ln(x).
You'll see a lot that log(x) is used instead of ln(x) in math books. Let's just say that it's mathematicians being weird

Things get even more awkward: when reading a computer science text, log(x) means $\log_2(x)$ most of the time: the logarithm with base 2! This makes sense again, because you never really need other logarithms.

What? That's just confusing...

Pengwuino
Gold Member

And your calculator may mean Log = $Log_{10}$ or it could be the natural log! In physics we NEVER mean base 10 or 2. And the thing is that you either are going to use base 10 or base 'e'. I've never see them mixed so text authors sometimes don't even bother explaining what Log means to them, you just have to pick it up on your own (not that it shouldn't be immediately obvious which is being used).

Like wlframalpha does it and my professor too.

My professor dosen't say "the natural log of x", he just says 'log x'

Why??

In high school they tell you log is log base 10, and ln is log base e. I suppose there's a mix of the two in calculus, depending on the instructor. But once you get onto the math major track, the natural log is the only log you ever care about, so you just call it log. I don't know what the conventions are for physicists and engineers, but for undergrad math majors and up, log means natural log.

Math people don't draw little arrows over vectors either

Math people don't draw little arrows over vectors either

Haha.

What's a physicist's definition of a vector space?

Ans: A set V such that if v is in V, v has a little arrow drawn over it.

In the book I Want to Be a Mathematician: An Automathography by Paul Halmos, he discusses this and other things in a section where he is discussing the way that mathematicians talk. From his POV, the lnx thing just spontaneously appeared in calculus text books.

Another thing he discusses is the fact that most math people say "minus 5" instead of "negative 5". This has been my experience, as well. Many of my profs. say "minus" instead of "negative."

^ I actually always say "negative" to describe a negative number. "Minus" is a verb to me, e.g. "three minus two".

About log ... I think "ln x" just looks prettier than "log x" when written. But as long as the paper is unambiguous, i.e. it is particularly distinguished that "log" indeed means natural logarithm or otherwise, there's no problem.

chiro

Like wlframalpha does it and my professor too.

My professor dosen't say "the natural log of x", he just says 'log x'

Why??

I think most applications deal with the natural base, so from this the natural base is assumed.

Interestingly enough though, I remember sources in the past used to refer to "log" x of being base 10 and "ln" x being the natural base, and I think many calculators use this convention as well.

^ I actually always say "negative" to describe a negative number. "Minus" is a verb to me, e.g. "three minus two".

About log ... I think "ln x" just looks prettier than "log x" when written. But as long as the paper is unambiguous, i.e. it is particularly distinguished that "log" indeed means natural logarithm or otherwise, there's no problem.

I think the reason that math people tend to say "minus" is that it is more general than "negative." For example, if we are working in a group (or ring or field) each element a has an inverse denoted as "-a". Now, groups aren't necessarily ordered so it doesn't make much sense to call this "negative a." So, I guess math guys just start "minus" for everything.

But, I could be wrong.

I thought people say "minus #" because they are foreign lol

I thought "negative #" was an Americanism, I've always said minus (as an adjective as well as a verb), and so have all my teachers/lecturers as far as I can recall.

Well, log(x) is always assumed to be a natural logarithm, unless otherwise stated. It is rarely used in base 10. On a calculator, log(x) has been designated as base 10 so that functions are not duplicated. However, both log(x) and ln(x) has been around long before calculators.

So here's the question - why have the ln(x) term?

So here's the question - why have the ln(x) term?

Because no matter who you are talking to, the only logarithm it refers to is the natural logarithm.

Well, log(x) is always assumed to be a natural logarithm, unless otherwise stated. It is rarely used in base 10.
I don't think this is true. On what do you base these statements? In my experience (which coincides with most of the answers above), most engineers and scientists take log to mean the common log, i.e., base 10. It is certainly not true that "log(x) is always assumed to be a natural logarithm, unless otherwise stated".

Historically, common logs were used far more widely than natural logs. Before computers and calculators were widely available it was harder to multiply numbers than to add. logs were used as a calculation aid to turn multiplication into addition. It was usual for science and engineering books to include log tables in the back for this purpose. You need common logs for this, so that you can easily find the log of 735, 735000, or 7.35 with the same table. (A slide rule works on the same principle, but it's hard to get more than 2-3 sig figs from a slide rule.) Natural logs, in contrast were an obscure mathematical curiosity to most of the world.

uart

Well, log(x) is always assumed to be a natural logarithm, unless otherwise stated. It is rarely used in base 10. On a calculator, log(x) has been designated as base 10 so that functions are not duplicated. However, both log(x) and ln(x) has been around long before calculators. So here's the question - why have the ln(x) term?
Historically log base10 was used far more than it is today. In the pre-computer era base 10 was unrivalled for tabulating (look up tables) and computational purposes. This is why log base 10 was called the common logarithm and, outside of pure mathematics, was just about universally referred to as "log". So traditionally "log" = "log10" was very commonly used and hence "ln" used for the natural log.

Today the applications for log base 10 are much diminished. We still use them for dB calculations and for some graphing/charting applications, but to be honest I cant think of a lot else that base 10 is now used for. So it's not surprising that in recent times the use of "log" to represent natural log has become more common, with log base 10 tending to get usurped.

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uart

LOL Me and pmsr3 posted almost exactly the same thing at almost exactly the same time. :)

And the thing is that you either are going to use base 10 or base 'e'. I've never see them mixed...
In just about every Neuroscience text in the world you will see something like the following:

$$\begin{eqnarray*} E & = & \frac{RT}{nF}\ln\frac{[A]_o}{[A]_i} \\ & = & 59\log\frac{[A]_o}{[A]_i} \mbox{mV for a univalent ion at physiological temperature} \end{eqnarray*}$$

That's the Nernst equation, first using the natural log, then the common log. The second form is useful because it's easy to remember that every factor of 10 gives you 60 mV.

I just say log ( x ) for ln ( x ), although I know that "historically" log( x ) is supposed to mean log in base 10. But hey, all logs are more or less the same thing.

I believe that computer scientists use the notation lg(x) for log in base 2

I like Serena
Homework Helper

I think the reason that math people tend to say "minus" is that it is more general than "negative." For example, if we are working in a group (or ring or field) each element a has an inverse denoted as "-a". Now, groups aren't necessarily ordered so it doesn't make much sense to call this "negative a." So, I guess math guys just start "minus" for everything.

But, I could be wrong.

I think that "we math people" (counting myself as one in this context) tend to call the inverse in a group denoted as "-a" as "the (additive) inverse of a" or "a-inverse" for short.
That is, we call it "inverse" to avoid ambiguity with anything that is not an "inverse".

Math guys would (or should) always opt for the terminology that avoids ambiguity at all costs, certainly within the field we're currently working in.
It's engineering guys and other people who tend to be sloppy with terminology.

Claude Bile

Base 10 logarithms are still prevalent in physics, particularly when trying to express many orders of magnitude graphically and in logarithmic units etc.

Back in the day, most physicists were mathematicians and vice-versa, hence the cross-pollination of little notational quirks like this.

Claude.

I think that "we math people" (counting myself as one in this context) tend to call the inverse in a group denoted as "-a" as "the (additive) inverse of a" or "a-inverse" for short.
That is, we call it "inverse" to avoid ambiguity with anything that is not an "inverse".

Math guys would (or should) always opt for the terminology that avoids ambiguity at all costs, certainly within the field we're currently working in.
It's engineering guys and other people who tend to be sloppy with terminology.

That hasn't been my experience. The profs I have usually refer to the additive inverse of a as "minus a". I can't remember a prof I have had since calculus that would say "negative a" instead of "minus a".

AlephZero
Homework Helper

I thought "negative #" was an Americanism, I've always said minus (as an adjective as well as a verb), and so have all my teachers/lecturers as far as I can recall.

In British English, you say "negative numbers" (as a general concept) but a specific number like -12 is "minus twelve" not "negative twelve".

It's farily obvious what "Minus numbers" means, but that is Globish, not English.

As a Brit, "negative twelve" sounds like the same sort of pedantry as saying "the measure of angle ABC = 45 degrees" rather than just "angle ABC = 45 degrees".

Back to the question of the logs, I'm not really sure about the notation, but your professor calls it the "log x" instead of "the natural log of x" because it's obviously easier to say.

Hurkyl
Staff Emeritus