inspired by a comment in this thread:
one poster says that a logarithm can’t have any units. I am used to that idea but mostly because my usual logarithm represents decibels.
In general, why shouldn’t one say that a logarithm has units?
I can say 100 seconds. units seconds.
I can say log10(100) = 2. units on the left are seconds. At least there is nothing to remove the units on the left. so shouldn’t be units on the right be seconds?
thanks
If the units on the right is seconds, then:
log(100 s) = 2 s
But in no sense is it actually two seconds.
a decibel is a unit defined as the results of a logarithmic formula, you don’t start with decibels, take the logarithm and get decibels. You start with a unit-less ratio and define a name for the result.
Hint:
Suppose X has units of seconds. What are the units of X + X^2?
The log function is similar to the above example. It doesn’t make sense to take the logarithm of a unit-ful quantity. In your example, what does your equation mean? When you take a logarithm of 100 seconds, what do you expect the 2 on the right hand side to represent, physically?
However, suppose X is 3 meters. X^2 would then be 9m^2, i.e. 9 square meters, which would be the area of a square with a side length of 3 meters.
Now, if you came up with 3 meters + 9 square meters, you have to think to yourself, “what does that mean in the real world?” because when you are running numbers or equations like that with real world units, usually you are doing so because the numbers and equations represent a useful model of something in the real world. If the result does not also represent something in the real world, then the utility of your model is in question.
Logarithms are exponents. For example log(10^x) = x.
You can get a logarithm with “units” if you can find a usage of A^x where x is not a pure number, but I don’t think you can.
I think log(100 seconds) = 2 + log(seconds).
Seeing the absurdity of " log(seconds)" helps answer OP’s question.
I’ve underlined “unit-less ratio” since this is key to how logarithms are used. In the example “2 + log(seconds)”, the 2nd term would be discarded and you’re working with the ratio of a a time duration to unit time.
The Richter scale, like decibels, is logarithmic and people may speak of “Richters” as though they were units. Perhaps one way to distinguish normal units from “logarithmic units” is to ask about zero. “Zero on the Richter scale” does not mean the ground is motionless. Instead, from Wikipedia
The principle to be obeyed is that scaling units on the left hand side of the equation and then performing the action on the scaled numbers should have the same effect as scaling the units on the right side of the equation.
As a toy example, if you have a 3m x 10m room, you can use the area formula l*w = A as:
3m * 10m = 30m^2
If you decide that you would rather have used centimetres as your length scale, you can do two things:
[ol]
[li]You can note that there are 10000cm^2 per m^2, and convert the right hand side by 30m^2 = 300000cm^2.[/li][li]You can note that there are 100cm per m, and rewrite the left hand side as 300cm * 1000cm, and then apply the formula to get the right hand side of 300000cm^2[/li][/ol]
And in both cases you get the same result.
Contrast this with the case of equation of the form log10(t) = x, where t is a time, and x has undetermined units. If we decide to measure t in seconds, and we find that t = 60, we get x = log10(60) ~= 1.778 of whatever units x is in.
But if we later think we should measure t in minutes, we now get x = log10(1min) = 0.
So no matter what units you are proposing x to be in, there is no unit conversion that converts 0 to 1.778, so no sensible units for x can be defined.
That’s why in any “real” physical law, if there is a log (or an exp, or any higher function) its arguments are always dimensionless combinations of parameters or constants.
In general, multiplying something that has units by a dimensionless number gives a result that still has the same units. But multiplying a log by an integer n gives a result equal to the log of the nth power of the original argument. If there was a unit in there, raising it to a power changes the units (length to the third power becomes volume for example). Multiplying a log by a non integer would change the units a not-very-sensible power law relation to a real unit (what do you do with feet^1.7723?).
The only use of logarithms I am aware of are fake units of convenience to normalize results to some agreed-upon standard. For instance, dB (or dBm), pH, or absorbance are all values normalized to some arbitrary “standard value” then given in logs because otherwise the numbers are or could be crazy large or crazy small.
“units” is a term which is used to label various kinds of additive data. If you have various quantities with the same “units”, then you know it is sensible to add them together to get another quantity with the same “units”.
But logarithms aren’t applied to additive data. Logarithms are applied to multiplicative data; their whole purpose is to turn multiplicative data into additive data.
If you have some datatype where multiplication isn’t meaningful, then logarithms aren’t meaningful either (in the same sense in which, say, addition of Fahrenheit temperatures isn’t meaningful).
If it does make sense to multiply, then it makes sense to apply logarithms, and the result will have some “units”; the “unit” will amount to a description of the base of the logarithm.
thanks all!
I am still working through the answers. I understand the ideas, but I still can’t quite reconcile them operationally.
what are the units of log10(10) = 1? why not the same units as 10? The 1 is as unambiguously the same number as A(hex) or 12(octal).
I have a number 10 with units of seconds. I can choose to describe that number in various ways. A. 12. 10. 1. They are all unambiguous once the definition is given. They all have the same units since they all represent the same entity.
Now to reconcile this with all these excellent answers.
They are both unitless, so they are the same units.
This is something completely different, and not a question of units. Here, you’re talking about changing the base of the number system you’re using.
In decimal, log10(10) = 1. Actually, that’s true in any base, as long as you use the same base for both "10"s. So in hexadecimal, log10(10) = 1, but also logA(A) = 1.
More generally, for any number X, logX(X) = 1, regardless of the base.
But none of that has to do with units.
To find absorbance, you calculate the log of the ratio of the intensity of the sample versus a reference intensity. Since you divide intensity by intensity, the resulting ratio is unitless. It’s not the logarithm that does this, the ratio was unitless before you applied the log. The logarithm is just a convenience.
It’s not that logarithms don’t have units per se, it is that
And why are natural logarithms “natural”?
Because the natural logarithm is uniquely the one with derivative 1/x, and uniquely the one that is the inverse of the exponential function (which itself is natural on account of its simple power series).
Long story short is that there is a lot of math that if you do using the natural logarithm comes out nice and simple with rational coefficients, but if you do using a different logarithm, comes out with all sorts of strange irrational constants of proportionality all over the place.
I believe I understand now. My point was that 10 = A = 12, just different ways of writing the number 10. 1=log10(10) however really isn’t another way to write 10. So my argument doesn’t hold up.
As has been said, one never takes the log of a dimensioned number. One always divides by a scale factor to produce a dimensionless number. I suspect there are good mathmatical reasons, but the simplest reason is simply that by doing so we avoid having to deal with units.
The “natural logarithm” of b can be defined as as the initial rate of growth of exponentiation with base b.
The function so-defined is of ubiquitous significance in calculus (arising, quite naturally, whenever one is interested in the rate of growth of an exponential function), and happens also to behave like a logarithm; thus, we call it the natural logarithm.
One can then go on to define its inverse function as the “natural exponential”, the value of the natural exponential on input 1 as the constant “e”, and so on.
In fact, dB is a ratio not a value. dBm is a power ratio related to 1 mW, so does have a value.
There are a bunch of different ways you can introduce the natural exponential and natural log functions. You could introduce general exponentials and logs, and then define the natural ones as the ones having slope of 1 where they cross the axis, and then prove that the natural log is the integral of 1/x. You could define the natural log in terms of the integral of 1/x, and then define the exponential as its inverse. You could define either one in terms of its power series, and work from there. You can define the exponential function as the function which is its own first derivative. Et cetera, et cetera.
If this is in response to my post, I agree, of course.
However, I would claim that most of the significance of the natural logarithm in calculus is ultimately in its relationship to the derivative of exponentiation [that the partial derivative of b^p with respect to p is ln(b) b^p, or, in other words, that b^p = 1 + ln(b) p, to a linear approximation in p around 0, or, in other words, that y(p) = b^p satisfies the differential equation dy/dp = ln(b) y].
This is equivalent to every other which way of defining it, tautologically, but for most typical calculus applications, I would argue this in particular to be the most salient fundamental property [e.g., the particular power series one could use to give a definition are only typically interesting because they reflect the above property, rather than the other way around].
Kiwi Fruit is correct that [deci]bels on their own are a unitless reference to a ratio. But to further confuse the dB issue perhaps the most common use of decibels is to refer to how loud something is (technically in power over area, known as sound pressure level). Those dBs are referred to a known value — 0dB is a specific power level and generally regarded as the threshold of human hearing. The additional qualifier is usually left off, but when confusion may occur ‘SPL’ is added to the quantity to clarify.
A common mistake in this regard : The volume indication on a stereo might give a reading in negative dB. Those are unitless: 0dB is the maximum power output. The actual dB SPL depends on the speaker used - something around 100 dB SPL would be reasonable for a home stereo at max power (0dB).