# “Minus” a logarithm

I have to explain in words this formula:

`A = -log(X)`

My guesses would be either:

1. A is the minus logarithm of X
2. A is minus the logarithm of X

Is one of these expressions, or yet another one, correct?

• @Catija negative is more for turning a single number into its negative counterpart than for entire expressions (as this logarithm). So you can say -9 is negative nine, but it doesn't work too well with other expressions. I'm not saying, however, that people won't understand what you mean :) – André Chalella May 19 '15 at 15:52
• You can have the "negative logarithm" of something... It's the equivalent of -1[log(x)] – Catija May 19 '15 at 15:53
• @AndréNeves It's math... -(-1) = 1... that's how math works. – Catija May 19 '15 at 17:40
• Do you want to speak the mathematical expression out loud, or write a verbal explanation of its meaning? – Ben Kovitz May 19 '15 at 20:01
• BTW, please allow at least a day or two before accepting an answer. For info about why this is helpful, please see “Not so fast! (When should I accept my answer?)”. Often the first answer is mistaken, which sparks discussion to improve it or write alternative answers, as I think happened today. – Ben Kovitz May 19 '15 at 21:37

After a long debate, it's clear that almost everyone normally writes in words `A = -log(X)` as either of:

A is minus the logarithm of X.

A is (the) negative the logarithm of X.

There is also evidence that some people use one exclusively over the other. It looks like British English favors the former (minus), while American English favors the latter (negative).

The following are also used, albeit not formally accurate:

A is minus logarithm of X

A is (the) negative logarithm of X

The reason for the inaccuracy is that there is only one logarithm, not one positive and one negative from which you could choose. However, this usage seems okay in some contexts, especially those in which logarithms are multiplied by -1 all the time (Chemistry's pH, for instance).

In spoken language, the same is true, often shortening the logarithm of to simply log and the like.

In order to avoid giving the impression that the final result of `-log(X)` is negative, however, some (not many) authors will use:

A is the opposite of the logarithm of X

This is much less popular than the other choices, and some people may mistake its meaning for `exp(X)` rather than `-log(X)`.

# Conclusion

Do use minus/negative the logarithm in general writing and speaking as it is much more current and understood by almost everyone, while also being formally accurate. In special contexts where the resulting sign issue might be a concern (perhaps elementary algebra), it may be more effective to address the possible misconception in an additional note than to resort to the opposite of form.

References for the unpopular opposite of variation:
http://en.wikipedia.org/wiki/Plus_and_minus_signs#Minus_sign

• Thanks, that was very clear. Indeed, I needed to put it in a formal contest. – Alessandro May 19 '15 at 17:33
• Just saying, when I asked about this in the Math chat room, the answer I got was that it should be "the negative" or "the opposite". I don't have any primary knowledge of this and I don't have enough math background to write an entire answer but... they seem to disagree with "minus"... – Catija May 19 '15 at 22:07
• Well, how about that! I just looked and Newton is indeed writing "oppoſite of the Logarithm" to mean –log n. It's interesting: he seems to regard the log of the numerator as going forward and the log of the denominator as going in the "oppoſite" direction, much like the habit of thought you pick up if you use a slide rule. – Ben Kovitz May 20 '15 at 0:14
• @AndréNeves That long dash is indeed a minus sign: arithmetic symbols seem to have become shorter over time. See the sample here, for example, for even crazier length. – David Richerby May 20 '15 at 6:57

## Any audience

With any mathematical audience, you can write:

A is the negative of the logarithm of X.

In speech, you can also say:

A is minus the logarithm of X.

A is the negative of log X.

A is minus log X.

## People who work with very small numbers

Some mathematically knowledgeable audiences work a lot with the logarithms of small fractions, but for convenience, they prefer to work with positive numbers. (The logarithm of a fraction is always negative.) If X is expected to be between 0.0 and 1.0, people in these fields would usually say:

A is the negative logarithm of X.

expecting that A will be positive. For example, in chemistry, you speak often of pH, where H is the concentration of available hydrogen ions in a solution—usually a very small fraction. An acidic solution with an H+ concentration of 10–4 is said to have a pH of 4. In fact, in chemistry, the p prefix is actually defined as a general mathematical operator with exactly this meaning:

In general, a lowercase "p" before a symbol means "negative logarithm of the symbol." (From Chemistry, by Whitten, Davis, Peck, and Stanley (2013), p. 713.)

You can find the term "negative logarithm" in almost any standard textbook explanation of pH.

People also become accustomed to saying "negative logarithm" when they work with information theory and statistics. In those fields, you often take the logarithms of probabilities. Since probabilities are usually fractions, this would result in negative numbers, so people customarily take "the negative log" so they can work with positive numbers. This Google Books search will show you lots of examples from fields outside chemistry.

## Algebraists

If you're talking with an algebraist, and you're making a point about the logarithm being a homomorphism from addition to multiplication, you could say:

A is the additive inverse of the logarithm of X.

but that would be very unusual. Normally you just say "negative".

In British English, "A is minus the logarithm of x" is fine; American English would, I think, prefer "negative" instead of "minus".

In British English, "minus" does not refer only to the operation of subtraction but also to what programming langauges would call the "unary minus" operation. Some examples of negative numbers are "minus one", "minus two" and so on. Since we're happy to take "two" to "minus two", we're happy to take "the logarithm of x" to "minus the logarithm of x". (Conversely, if you want to emphasize that a quantity is positive, you can say, for example, "plus three"; I'm not sure if Americans would prefer "positive three".)

"The minus logarithm of x" doesn't work because it suggests there's a thing called a "minus logarithm" and you're taking that function of x. On the other hand, the comments indicate that "the negative logarithm of x" is quite often used to mean "-log x". "Minus" and "negative" aren't quite synonyms: you'll notice that, above, I said "negative numbers"; "minus numbers" would be considered incorrect even in British English.

• +1 for constructiveness. May I only make explicit that "the negative logarithm of x" shouldn't be used as well, for the same reason regarding "the minus logarithm of x." – André Chalella May 19 '15 at 19:26
• @BenKovitz I can totally understand that. We probably agree it's a more or less loose usage, however. I think this question asks for a more formally correct form. On the other hand, I acknowledged the pervasiveness (and usefulness) of the construct you offered in the question comments. – André Chalella May 19 '15 at 19:43
• +1, esp. for the unary operator. Let me note that I do not like 'the negative logarithm' because to me that does sound like 'negative logarithm' is a function. – Keith May 20 '15 at 5:04
• @Keith The negative logarithm is a function. In some fields, it's convenient to call it by a special name since it occurs so frequently and is applied only to numbers in [0, 1]. (It's often extended so 0 is in the domain.) – Ben Kovitz May 20 '15 at 8:17

Certainly in a modern, British classroom, we would probably simply read this aloud as:

A equals minus log x

For us, it would not really be necessary to expand everything out. For example, I would never say "the logarithm of" - much too wordy. In fact, we even took notes on "laws of logs", not "laws of logarithms".

However, because of this tendency to abbreviate, stress, intonation and timing become very important. To indicate that log is a function acting on x (i.e. that log goes with x), we would say something like:

A equals minus (log x)

There is a small pause between the "minus" and the "log x" and "log x" is said in a bit of a rush.

This is something that can only come with practise and is practical in a classroom setting where people likely have the equation written out in front of them.

The take home message is that, certainly for me, I have always been taught to read mathematics using the short forms (I say log, not the logarithm), but I use pauses, stress and intonation to indicate what goes with what.

Consider another example:

a - (b + c)

a [pause] minus [pause] b plus c

With the italics indicating stress and b plus c pronounced all in one go.

Whereas:

a - b + c

Would just be:

a minus b plus c

N.B.: I don't know if you're a mathematician, but if you are, you should be aware that some people use the pronunciation "log" for the natural logarithm - log_e or ln - where others use it for the common logarithm - log_10/log/Log/lg. There is, then, a danger with simply saying "log", so you should know your audience. It may be necessary to specify the base you are using. We would say, in this case, "log to the base 10" for log_10.

• I'd read out "a-b+c" as "a minus b [pause] plus c". To me, that makes it clearer that I really do mean a-b+c and not a-(b+c). – David Richerby May 20 '15 at 8:25
• +1 for "minus log x"; that's certainly what I, as a mathematician, would use, unless I was speaking to an audience who might not know what a logarithm is (in which case I'd take the time to spell it out and to explain what it does in context, as in "A is the negative logarithm of x; that is, a is zero when x equals one, and gets larger and larger as x gets closer to zero [etc.]"). – Ilmari Karonen May 20 '15 at 16:37

You could say "A is equal to negative log X". Minus is generally refers to the operation of subtraction. Saying additive inverse is also perfectly correct, but rather cumbersome. Saying "A is equal to the opposite of the logarithm of X" is really imprecise, because opposite isn't a well-defined mathematical term. The "opposite" of log(X) could be any of -log(x), 1/log(x), or exp(X). Negative leaves no ambiguity about your meaning and is very concise.

• Sorry, but you're stretching a lot the intersection between opposite and inverse. No one will ever say opposite means 1/log(x), much less exp(x). No one calls the inverse of a function its opposite. Otherwise, would you be able to back that claim with references? – André Chalella May 19 '15 at 16:27
• Nobody, in my experience, calls anything the "opposite" of a function, number or anything else. – David Richerby May 19 '15 at 18:59
• @DavidRicherby At the grade school level, division is commonly referred to as the "opposite" of multiplication, and addition as the opposite of subtraction. (In the U.S. at least) For example: google.com/search?q=opposite+of+division ...no, I don't like it either, but it is out there. – Adam May 19 '15 at 21:03
• @Adam I agree, but that's informal (and arguably describing something as an opposite, not calling it). It also matches the theme of "opposite" being interpreted as inverse, rather than negation. – David Richerby May 19 '15 at 21:09
• To the best of my knowledge, "opposite" is not a recognized mathematical term that means anything specifically. If someone casually said to me, "the opposite of the log function", my first thought would be that he means e^x. I doubt I'd think of 1/log(x) but if he said that was what he meant, I wouldn't think it absurd. – Jay May 20 '15 at 13:19

If I was speaking casually, I'd read it "A is minus log x". If more formally, "A equals negative logarithm x" or "A equals the negative of the logarithm of x".

I probably wouldn't say "negative log x", without the word "of", because that makes it sound like there is a negative log and a positive log. Like there is a negative square root and a positive square root. On the other hand, if the audience knows anything about logs, they now that they don't come in pairs like that, so this would only be a valid concern if you were teaching what logarithms are or some such.

I'm sure some pedants would say that "minus" is a binary operation, i.e. you can talk about "x minus y", but you can't say "minus x", you must say "negative of x". But there's little gained from such a distinction.