# Logarithm of numbers with units

#### Repetit

Hey!

I have always learned that functions like logarithms, exponentials, trigonometrics etc. have to operatore on pure numbers and not numbers with units. For instance, you cannot write:

Sin ( 5 kg*m/s^2 )

But in chemistry I often find formulas where logarithmes of numbers with units are taken. If some of you happen to own Atkins - Physical Chemistry 8th edition you can look at page 807 and see what I mean. They take the logarithm of k and A which have units (the Arrhenius equation).

How can you do something like that?

Thanks!

#### LeonhardEuler

Gold Member
Are you sure you're looking at the units of the entire argument and not just one part? You can't have something like:
$$\ln{k}$$
But you can have
$$\ln{\frac{k_1}{k_2}}$$
because the units of k_1 cancel with those of k_2. You can also have something like:
$$k=k_0e^{\frac{-E}{RT}}$$
because E has units of energy/mol, R has energy/(temperature*mole) and T has temperature so all of the units cancel.

#### Repetit

Yes I know that, I see it all the time in physics. But that's not the case here. They are taking the logarithm of a number with units.

#### LeonhardEuler

Gold Member
Alright, I've found the seventh edition of the book and, if the equation here is the one you are talking about, then-yes, in the form he has it written, it is the log of a number with units, but just rearanging:
$$\ln{k}=\ln{A} -\ln{\frac{E_a}{RT}$$
$$\ln{\frac{k}{A}}=-\frac{E_a}{RT}$$
The important thing is that it doesn't matter what units you express "k" in, you'll just get "A" in those same units. What you want never to see is an equation like:
$$k=C\sin{\pi T}$$
With an equation like that, your answer will always depend on the units. If you measure T in Kelvin, then at 273K, k is 0 because you have sin of an integer times pi. But if you measure in rankine, then k is not 0 as long as C is not 0 because 273K=491.4R which is not an integer. You don't end up with contradictions like these in the equation he gives.

If you look at his graph, he has ln K on one axis. Personally I would not write it like this. Often you'll see authors write something like:
$$\ln{\frac{k}{k^*}$$
Where $k^*$ is a unit k. So if the units of k are $s^{-1}$, then $k^*=1s^{-1}$, which gives the same numerical values, but doesn't raise the objections you're talking about.

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#### matt grime

Homework Helper
Are people honestly being taught that you can't take logs of numbers if there is a unit attached? Of course you can. Whether that is then a meaningful quantity in itself is another matter entirely. If you indeed 'couldn't' take logs of numbers with units then what is the point of a log-log graph?

#### chroot

Staff Emeritus
Gold Member
The log of, say, some number of joules has units of "log joules." The unit "log joules" may not be physically meaningful, as matt grime points out, but that doesn't mean it's not a valid unit.

You must also realize that all numbers are ratios, because all numbers x can be expressed as x / 1.

- Warren

#### LeonhardEuler

Gold Member
Are people honestly being taught that you can't take logs of numbers if there is a unit attached? Of course you can. Whether that is then a meaningful quantity in itself is another matter entirely. If you indeed 'couldn't' take logs of numbers with units then what is the point of a log-log graph?
It's true that its mathematically possible to deal with logs of units, but its very useful to avoid them. Whenever you are doing a lot of manipulations, you can always tell you've done something wrong if you end up with units inside the argument of a log or trig function etc. It happens all of the time. There may be other situations where it is useful to be aware that this is not necessary, but I have not encountered any.

#### symbolipoint

Homework Helper
Gold Member
Logarithms of numbers are useful as a method to treat data. Sometimes powers of numbers are useful as a method to treat data. Logarithms or powers often allow us to give an easy (or easier) description to numerical relationships. Recall, someone mentioned log-log graphs, and semi-log graphs.

#### LeonhardEuler

Gold Member
Yeah, but when you make a graph you can always do the $$\ln{\frac{k}{k^*}$$ thing to make the argument of the log dimensionless. I'm not denying the usefulness of logs, just saying that I have never encountered a situation where it was useful to use logs with units.

#### matt grime

Homework Helper
It's true that its mathematically possible to deal with logs of units, but its very useful to avoid them. Whenever you are doing a lot of manipulations, you can always tell you've done something wrong if you end up with units inside the argument of a log or trig function etc. It happens all of the time. There may be other situations where it is useful to be aware that this is not necessary, but I have not encountered any.
We're all aware of dimensional analysis. But it is plain wrong to say that taking logs or sins of numbers with dimensions is not allowed.

#### LeonhardEuler

Gold Member
We're all aware of dimensional analysis. But it is plain wrong to say that taking logs or sins of numbers with dimensions is not allowed.
What I meant was that equations like
$$k=C\sin{\pi T}$$
are non-physical and it should raise a red flag if you see one in P-chem.

#### Gib Z

Homework Helper
When applying it to Physics, log units can still be somewhat useful, just like the imaginary unit. The solution to a certain section of an equation may give you a log unit or complex number, but it can turn out it becomes exponentiated or multiplied with another complex number to get us back to happy units :D. log units could also be useful for calculations as Logarithms as they reduce multiplication to addition, and division to subtraction.