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

For the discussion of math. Duh.

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

Note- This seems like the thing that there would probably be another older thread on, but my searching results turned out to be inconclusive. If this is a repeat post, I apologize.

After getting in a multitude of debacles on this, I have decided to seek our the XKCD fora for their views. So, I ask you all: to denote a logarithm to the base e, which to you prefer: log(x), or ln(x)? Personally, I write log(x), in fact, I cringe whenever I see ln(x). This is mostly because I have never ever seen a log to the base 10 used anywhere, so having log(x) mean base 10 by default is pretty silly. It also looks much more elegant to use log(x) as base e, and logb(x) as a logarithm to the base b. What do you guys think?

B.Good
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### Re: ln(x) or log(x)?

I'm the exact opposite of you, I cringe every time I see log(x) to denote ln(x), it has been drilled into my head to mean "log base 10". Although log base 10 is rare they are used in chemistry to determine the pH of a substance, and I would bet that there are other uses though I am unfamiliar with them, granted, log base e is far more common. Also, I think ln(x) looks much nicer than log(x) and it takes less time to write than log base e, or even log for that matter.

greeniguana00
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### Re: ln(x) or log(x)?

In middle school, I was taught log(x) in terms of base 10. Only later did I learn about e and its significance.

We were taught that was because it keeps the numbers in calculations nice, and helps you build an intuition for it. For example:

10^0=1
10^1=10
10^2=100
10^3=1000
...
10^k = 100...000 (k zeros)

Then you can approximate (within +/-1) the logarithm of any number in base 10.

For example: What is log(723590943)?

If log is taken base 10, then we know it's between 8 and 9.

More important, it's easy to see that the rules of logarithm work:

3 = log(1000) = log(10*10*10) = log(10)+log(10)+log(10) = 1+1+1 = 3.

If we have a logarithm table for 0 up to 10, we can use these rules to calculate log for for any positive number, no matter how large! Example:
log(723590943645) = log(7.23590943645*10^11) = log(7.23590943645)*log(10^11) = 11*log(7.23590943)

If you work with log base e, you don't have all these nice properties, and the intuition is less clear when learning it for the first time.
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Mike_Bson
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### Re: ln(x) or log(x)?

greeniguana00 wrote:If you work with log base e, you don't have all these nice properties, and the intuition is less clear when learning it for the first time.

Or just write log10(x) when teaching kids about logarithms to build this intuition. In fact, that seems like such a trivial matter, it shouldn't have anything to do with this, and I doubt it does.

greeniguana00
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### Re: ln(x) or log(x)?

Mike_Bson wrote:
greeniguana00 wrote:If you work with log base e, you don't have all these nice properties, and the intuition is less clear when learning it for the first time.

Or just write log10(x) when teaching kids about logarithms to build this intuition. In fact, that seems like such a trivial matter, it shouldn't have anything to do with this, and I doubt it does.

In complex analysis, the complex logarithm is defined using the real logarithm, i.e. Log z := ln r + iθ = ln | z | + iArg z

So it makes sense to use ln for the (more restricted) real logarithm, and log for the extended complex logarithm.

e is really only beautiful in complex analysis anyway.

Then there is also no problem is defining log to be base 10 as people are introduced to it.

Using log to mean base 10 (except in complex) and ln to mean natural log is the most unambiguous way to do it.
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Mike_Bson
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### Re: ln(x) or log(x)?

greeniguana00 wrote:In complex analysis, the complex logarithm is defined using the real logarithm, i.e. Log z := ln r + iθ = ln | z | + iArg z

I am a noob to complex analysis. Would you kindly inform me the difference of saying Log(z) and ln(r)? Like, I know the former is the log of a complex number, and the latter real, but why use different notation for them? Doesn't it make sense to say ln(z) = ln(r) + iθ +2πik, if we're using ln(x) as the natural logarithm of x?
Using log to mean base 10 (except in complex) and ln to mean natural log is the most unambiguous way to do it.
I disagree. With everyone disagreeing on the definition, log(x) is definitely not unambiguous. Every textbook I've seen, it used ln(x) as the natural log (most likely to cater to non-mathematicians), and log10(x) as the base 10 log.

skullturf
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### Re: ln(x) or log(x)?

My personal preference is for log(x) to mean the natural log. But when teaching calculus, I defer to all the textbooks, and use ln.

In a more specialized academic context -- writing a research article, or giving a presentation to fellow PhDs -- I would use "log" to mean the natural log. (Possibly I would include a brief remark to that effect at the beginning.) I should point out that I'm in math (probability, classical analysis, and such) -- chemists or earth scientists, say, might understandably have a different convention.

Two aesthetic flaws that "ln" has, in my opinion:

-- The lower case L looks like a capital i. "ln" looks like "In". I've noticed that students, especially some whose native language doesn't use our alphabet, make the mistake of writing it as "In".

-- How do you pronounce "ln"? I grew up calling it "lawn" or "lon" (I grew up in a place where "don" and "dawn" are homophones, which isn't true everywhere). I've also heard "lin" and "ell en". Lately, I prefer "ell en", but I think all the possibilities sound a little awkward or dorky. (I realize personal aesthetics are subjective, though.)

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Mo' Money
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### Re: ln(x) or log(x)?

I pronounce it "log."

I pretty much write "log" for the natural log, except when teaching non-math people, when I use "ln" because they seem to like it that way. (Also, the TI calculators say "LN" on the log button and "LOG" on the log10 button, so if I use "log" when teaching I find that the students are likely to push the wrong button at some point and get confused.)

If I have to use a log base 2, I would just write that log2 ("lg" is an abomination because it is not pronounced differently from "log"). I almost never have to use log base 10 or anything else, but I'd do those the same way.

greeniguana00
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### Re: ln(x) or log(x)?

Mike_Bson wrote:
greeniguana00 wrote:In complex analysis, the complex logarithm is defined using the real logarithm, i.e. Log z := ln r + iθ = ln | z | + iArg z

I am a noob to complex analysis. Would you kindly inform me the difference of saying Log(z) and ln(r)? Like, I know the former is the log of a complex number, and the latter real, but why use different notation for them? Doesn't it make sense to say ln(z) = ln(r) + iθ +2πik, if we're using ln(x) as the natural logarithm of x?

Well, first, there is a difference between "log(z)" and "Log(z)". The first is single-valued, but doesn't vary continuously on the complex plane. The second is multi-valued, but can be made to vary continuously.
This difference arises from arg(z) vs. Arg(z). Both measure the "angle" of a complex number, but the angle is really only unique up to multiples of 2pi. (i.e. pi/2 refers to the same angle as pi/2 + 2pi). arg(z) gives you all the numbers that refer to that particular angle, while Arg(z) returns only the number between -pi and pi.

Using "log" to refer to base e is much more natural in the complex numbers, because e is so tied up in the algebra of the complex plane. For example, e^(i*t) = cost + i*sint. Specifically, e^(i*pi) = -1. No one would ever use log base 10 in the complex numbers.

e doesn't really have algebraic properties that interesting in the real numbers, so the natural log and base 10 log both have potential uses. Seeing as both ln and log are used when dealing with real numbers, and seeing as there are people who might use both frequently, it's probably good for there to be an easy distinction between them that doesn't require tedious subscripts.
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achan1058
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### Re: ln(x) or log(x)?

For general purposes, I use log for base 10, ln for base e, and lg for base 2. For field specific purposes, I tend to use log for all, letting people infer from context. Besides, for things like analysis and complexity theory, which base of log often don't matter that much anyways.

Jyrki
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### Re: ln(x) or log(x)?

This is something that we simply have to agree to disagree on. It depends on the field (i.e. context), which base is the most common. The weight of history also comes into play. As do the choices made by the authors of most common textbooks and computer algebra systems.

My personal preference is to always use 'log' with a subscript, 'lg' for base 10, 'ln' for base e, and 'lb' for base 2 (though I might also write log2 for that). But this is just a preference.

If I'm coerced to make a pick, I would prefer 'log' to mean base 10. Admittedly that is because during my formative years calculators where not at all common, so folks learned to work with logarithm tables well before they learned anything about derivatives, and at that point base 10 was the obvious choice. In addition to the pH-scale, base 10 is used in telecommunications, because both error probabilities and signal-to-noise ratios are often described in powers of ten. The former because it is then easy to quickly come up with ball park figures, of how often something goes wrong - the latter because it is measured using the decibel (dB) scale.

Of course, in the context of complex analysis 'log' surely means base e, or rather, (one of the branches of) the inverse of the complex exponential function.

I frankly don't see the point of trying to compress the names of these functions in spoken language. There is usually enough background noise, and my goal is then to minimize the chance of anyone mishearing it. May be my bias to lecture room setting shows here ? I would simply use phrases like "natural log(arithm)", "Briggs' log" or "log base 10" and "binary log" or "log base 2". In particular when to the chalkboard. The audience immediately also gets an idea of my preferences making communication in the immediate future that much less prone to errors.

That was the tl;dr; version. The summary: it all depends on the context.

forgetful functor
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### Re: ln(x) or log(x)?

Once you get past introductory calculus classes, there is pretty much universal agreement in math that log means the "natural" or "base-e" logarithm. In fact, I don't recall ever seeing "ln" in any math textbook beyond the calculus level.

314man
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### Re: ln(x) or log(x)?

I've always learned that ln is base e and log is base 10. I see that almost 100% of the time. I notice my current (well the final was today so not current anymore) calc prof says "log" for the natural log. I'm only a 2nd year university student so I don't know if the textbooks and profs start switching to log.

Personally I think it should be as how I see it because almost everyone first learns logarithms in base 10, and it's hard for people to lose their habits. It's especially important for teachers because they throw around 'log' and it takes students a while to get that he/she means the natural log.

Also ln is shorter to write than log which is never a bad thing haha

NathanielJ
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### Re: ln(x) or log(x)?

And I use ln(x) for base e since that's just what I see most frequently used.
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### Re: ln(x) or log(x)?

I usually prefer ln(x) as logarithm to the base e and log(x) to the base 10.

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

In my experience mathematicians seem to prefer log and physicists seem to prefer ln. Theoretical physicists (being closer to mathematicians) seem divided on the issue. But certainly none of them would ever mistake log(x) for log10(x). And no mathematician would fail to understand the meaning of ln(x). So there's not really a problem.

I myself use log. I actually consciously made that choice as a way of saying "I may be a physicist, but I do enjoy mathematics a lot".
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Mike_Bson
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### Re: ln(x) or log(x)?

Diadem wrote:I actually consciously made that choice as a way of saying "I may be a physicist, but I do enjoy mathematics a lot".

I said something like, ''I don't give a damn about base 10, so I won't dignify it by saying ln.''

Eastwinn
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### Re: ln(x) or log(x)?

I alternate between them.
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Wnderer
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### Re: ln(x) or log(x)?

That's what's wrong with you kids today. Saying log(x) has base e. Haven't you heard of sliderules and log-log and semilog graph paper or decibels? Let say you only have regular graph paper to graph your log curve. Well if you know that the log(2) ~= 0.3, and the log(pi) ~=0.5, you can make your graph. So for every 10 ticks, 1 at 0, 2 at 3, 4 at 6, 8 at 9 and pi at 5. Just scale them by tens for what ever scale you need. No you kids need a super computer to do anything.

david.lewis314
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### Re: ln(x) or log(x)?

When studying calculus in high school we wrote ln for natural logarithm. However, while doing maths at university we tended to use log to indicate the natural logarithm, although some lecturers still preferred to write ln. I think that if you wrote log in the context of a maths paper, people would assume you meant the natural logarithm, but ln would still be understood. I did get the impression that some people regarded writing ln as something not done by 'real mathematicians', but that was just an impression.

Tetra
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### Re: ln(x) or log(x)?

Student in my class: So you're using "log" for the natural log?
Physics professor: I don't know of any other kind.

phlip
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### Re: ln(x) or log(x)?

I think it's similar to degrees/radians... when first teaching trig, everything's done in degrees, because that's what the students would be familiar with. Trying to introduce radians at this point would just result in a lot of "but... why?". But then when you get to calculus, or complex numbers, the advantages of radians become apparent. People will still use degrees over radians colloquially, and in fields other than pure maths (physics et al) but within maths, it's typically assumed to be radians.

Similarly for logs... when first teaching them, everything's in decimal, 'cause that's what the students are familiar with. But later, in calculus, the value of e is actually relevant, and the natural log is useful.

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

Wnderer wrote:That's what's wrong with you kids today. Saying log(x) has base e. Haven't you heard of sliderules and log-log and semilog graph paper or decibels? Let say you only have regular graph paper to graph your log curve. Well if you know that the log(2) ~= 0.3, and the log(pi) ~=0.5, you can make your graph. So for every 10 ticks, 1 at 0, 2 at 3, 4 at 6, 8 at 9 and pi at 5. Just scale them by tens for what ever scale you need. No you kids need a super computer to do anything.

Wait. Why would I want to do something as crazy as actually calculating a result in the first place? What do you take me for? An experimental physicist?
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### Re: ln(x) or log(x)?

log is base e, because I want to be differentiating. and having bases other than e makes my life more complicated. I don't think you'll find a statistician who'll do it any other way.
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314man
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### Re: ln(x) or log(x)?

phlip wrote:I think it's similar to degrees/radians... when first teaching trig, everything's done in degrees, because that's what the students would be familiar with. Trying to introduce radians at this point would just result in a lot of "but... why?". But then when you get to calculus, or complex numbers, the advantages of radians become apparent. People will still use degrees over radians colloquially, and in fields other than pure maths (physics et al) but within maths, it's typically assumed to be radians.

Similarly for logs... when first teaching them, everything's in decimal, 'cause that's what the students are familiar with. But later, in calculus, the value of e is actually relevant, and the natural log is useful.

I get what you're saying, although there is a difference. Degrees/Radians actually gives different answers when calculating. When doing trig functions by itself, there is no real clear advantage. But if you have a function like xsinx, there is a huge difference between degrees and radians. On the other hand, there is no difference between using ln and log (in base e of course) other than how it looks visually or pronounced

phlip
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### Re: ln(x) or log(x)?

I wasn't presenting it as a difference between "log(x)" (implied base e) and "ln(x)", but as a difference between "log(x)" (implied base e) and "log(x)" (implied base 10).

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Mike_Bson
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### Re: ln(x) or log(x)?

phlip wrote:I think it's similar to degrees/radians... when first teaching trig, everything's done in degrees, because that's what the students would be familiar with. Trying to introduce radians at this point would just result in a lot of "but... why?". But then when you get to calculus, or complex numbers, the advantages of radians become apparent. People will still use degrees over radians colloquially, and in fields other than pure maths (physics et al) but within maths, it's typically assumed to be radians.

Similarly for logs... when first teaching them, everything's in decimal, 'cause that's what the students are familiar with. But later, in calculus, the value of e is actually relevant, and the natural log is useful.

Plus ln notation and degrees are also similar in the sense that neither of them should ever be used .
Last edited by Mike_Bson on Wed Dec 22, 2010 4:27 am UTC, edited 1 time in total.

Dopefish
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### Re: ln(x) or log(x)?

I like ln(x) myself. Less writing, and I don't think theres any amibiguity like with log(x).

If I encounter log(x), if I don't automatically know from context (which I usually do), I'll either plug in log(10) to see if I get 1 or some decimal (when I encounter it within a program for example), or if it's in a book I'll hunt around until I can find an example where it's clearly base e (or base 10), and use that as that standard for the remainder of the book.

314man
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### Re: ln(x) or log(x)?

phlip wrote:I wasn't presenting it as a difference between "log(x)" (implied base e) and "ln(x)", but as a difference between "log(x)" (implied base e) and "log(x)" (implied base 10).

capefeather
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### Re: ln(x) or log(x)?

I don't think that notations should be personally adopted on the basis of some kind of elitist mentality, as it seems to be in this case. I honestly prefer ln just because it's less ambiguous, but when I write log I typically mean the natural log, too. I'm not all that consistent, and I tend to adopt the notations of whatever math profs I have at the time, but absolute consistency got overrated ever since we needed more than 52 constants. It doesn't really matter with ln/log, but with degrees/radians, it is often better to use degrees in non-math-related contexts because they're better for actually measuring, which is KIND OF important for applying our math to real world situations. It's "nice" in that context to have a full revolution represented by an algebraic number - a NATURAL number, even.

dag618
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### Re: ln(x) or log(x)?

I was always taught that log(x) is log base 10 of x and ln(x) is log base e of x. It would get confusing, especially to an engineer like myself, when it comes to things like http://en.wikipedia.org/wiki/Decibel since very few engineers will always specify the base of the log.

Nic the Man
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### Re: ln(x) or log(x)?

On the topic of engineers, I'd be curious to hear stories of some sort of construction failure due to someone down the line mistaking log 10 with log e. It would not surprise me.

Personally, I'm a log = 10, ln = e, lg = 2. But it's all very contextual. If I'm doing CS work, nearly every logarithm is base 2 so "log = lg = log2". Whereas in the mathematics realm, log = ln = loge. If I'm in the engineering realm, and (Much to the chagrin of some of my professors), ln.

Plus, I've always learned to hand-write the ln with a curly lower-case l just to differentiate it from the capital i. So I like curly.

Talith
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### Re: ln(x) or log(x)?

I wonder if the use of i instead of j for complex numbers has ever caused a transformer station to blow up or something... (cos if it hasn't then they should stop using j dammit! )

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

It doesn't really matter with ln/log, but with degrees/radians, it is often better to use degrees in non-math-related contexts because they're better for actually measuring, which is KIND OF important for applying our math to real world situations. It's "nice" in that context to have a full revolution represented by an algebraic number - a NATURAL number, even.

Yeah... ::sarcastic sigh:: You just calibrate your protractor to read from 0 to 2 (2=two half revolutions), then use [imath]\pi[/imath] the same way your would use [imath]^\circ[/imath] ---as a unit.

Actually, I prefer to measure angles so that the winding number is equal to the angle measure (angle of measure 1 is one full rotation). If I want finer resolution to my measurements, I am fully capable of using rational numbers. Dividing the circle into 360 parts seems so excessing, and [imath]2\pi[/imath] parts so messy (unless you are calculating arclength!), but letting a circle be unity---Ah! now that is beautiful.

Talith
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### Re: ln(x) or log(x)?

Well you're more likely to be working with a circle of radius 1 as opposed to a circle with circumference 1 and I'd much rather have 2pi radians circumference than have a circle of 0.5(pi)^-1 radius.

Eebster the Great
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### Re: ln(x) or log(x)?

There are plenty of practical applications of base ten logs for the obvious reason that we use a base ten number system. So it shows up in logarithmic scales like pH, sound intensity (bels), and the Richter's scale. There are plenty of practical and theoretical applications of base e logs for reasons obvious to anyone who has taken calculus.

Wouldn't it be great if we could distinguish between these on paper?

Reference two for why I use ln(x) for loge x: my TI calculator.

But in all seriousness, how often does this actually cause confusion? It seems like it should generally be pretty obvious from the context.

Oh, and I pronounce "ln(x)" exactly like that, "the el en of ex" (or just "el en ex"). Or "log" if I'm lazy .

E: Oh yeah, and replacing "i" with "j" for complex numbers is seriously confusing. My mind always does something like, "wait, this is a vector? No, no, that's not a unit vector and they're adding it to a scalar. But they never defined the variable, and there's no index . . . Oh shit, these aren't quaternions are they? I've never used those. No, that would be weird. Oh shit, they're electrical engineers." Some physicists do this too. Very annoying.

micawber
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### Re: ln(x) or log(x)?

The only time where I have ever seen this cause confusion in practice is in numerical calculations; bizarrely, Microsoft Excel uses log() for the base-10 log but VBA uses log() for the natural log. So switching back and forth from one to the other can cause errors.

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

Eebster the Great wrote:E: Oh yeah, and replacing "i" with "j" for complex numbers is seriously confusing. My mind always does something like, "wait, this is a vector? No, no, that's not a unit vector and they're adding it to a scalar. But they never defined the variable, and there's no index . . . Oh shit, these aren't quaternions are they? I've never used those. No, that would be weird. Oh shit, they're electrical engineers." Some physicists do this too. Very annoying.

Yeah, the whole j thing seems completely unnecessary (and something I don't use, as my knowledge of the subject comes form physics). Sure capital i is current, but i and I are pretty distinct. Besides that, in an electrical context, capital (vector) J I was taught to mean volume current density anyway, so theres still potential ambiguity, if not more so since j and J are probably easier to confuse imo. Never mind that J could be the unit joule, which also has potential to pop up in electrical problems (although I don't know what the engineers are doing).

Eebster the Great
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### Re: ln(x) or log(x)?

Dopefish wrote:
Eebster the Great wrote:E: Oh yeah, and replacing "i" with "j" for complex numbers is seriously confusing. My mind always does something like, "wait, this is a vector? No, no, that's not a unit vector and they're adding it to a scalar. But they never defined the variable, and there's no index . . . Oh shit, these aren't quaternions are they? I've never used those. No, that would be weird. Oh shit, they're electrical engineers." Some physicists do this too. Very annoying.

Yeah, the whole j thing seems completely unnecessary (and something I don't use, as my knowledge of the subject comes form physics). Sure capital i is current, but i and I are pretty distinct. Besides that, in an electrical context, capital (vector) J I was taught to mean volume current density anyway, so theres still potential ambiguity, if not more so since j and J are probably easier to confuse imo. Never mind that J could be the unit joule, which also has potential to pop up in electrical problems (although I don't know what the engineers are doing).

lowercase i is frequently used to represent current, though. Sometimes you will see capital I used as a constant and lowercase i used as a variable, or sometimes they will be used in different contexts, or sometimes i will be complex current and I will be the real part.

Yeah, it actually seems like EE are pretty big on ambiguity . . .

Oh, and j as the complex unit still isn't as annoying as the cgs system. God converting between systems is a nightmare.

kernelpanic
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### Re: ln(x) or log(x)?

I use log to mean base 10 and ln for e. This is because I use ln more commonly, and it's a bit shorter. (Yes, I'm that lazy)
I'm not disorganized. My room has a high entropy.
Bhelliom wrote:Don't forget that the cat probably knows EXACTLY what it is doing is is most likely just screwing with you. You know, for CAT SCIENCE!