Logarithm of numbers with units

In summary, the conversation discussed the use of logarithms, exponentials, and trigonometry with units in mathematical equations. While it is mathematically possible to use units with these functions, it is generally more useful to avoid them and make the argument of the function dimensionless. This can be achieved by rearranging the equation or using the log of a unitless ratio. The conversation also touched on dimensional analysis and the importance of avoiding non-physical equations in physical chemistry. Overall, the use of log units can be helpful in certain calculations, but it is important to be aware of their potential limitations.
  • #1
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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!
 
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  • #2
Are you sure you're looking at the units of the entire argument and not just one part? You can't have something like:
[tex]\ln{k}[/tex]
But you can have
[tex]\ln{\frac{k_1}{k_2}}[/tex]
because the units of k_1 cancel with those of k_2. You can also have something like:
[tex]k=k_0e^{\frac{-E}{RT}}[/tex]
because E has units of energy/mol, R has energy/(temperature*mole) and T has temperature so all of the units cancel.
 
  • #3
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.
 
  • #4
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:
[tex]\ln{k}=\ln{A} -\ln{\frac{E_a}{RT}[/tex]
[tex]\ln{\frac{k}{A}}=-\frac{E_a}{RT}[/tex]
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:
[tex]k=C\sin{\pi T}[/tex]
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:
[tex]\ln{\frac{k}{k^*}[/tex]
Where [itex]k^*[/itex] is a unit k. So if the units of k are [itex]s^{-1}[/itex], then [itex]k^*=1s^{-1}[/itex], which gives the same numerical values, but doesn't raise the objections you're talking about.
 
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  • #5
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?
 
  • #6
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
 
  • #7
matt grime said:
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.
 
  • #8
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.
 
  • #9
Yeah, but when you make a graph you can always do the [tex]\ln{\frac{k}{k^*}[/tex] 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.
 
  • #10
LeonhardEuler said:
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.
 
  • #11
matt grime said:
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
[tex]k=C\sin{\pi T}[/tex]
are non-physical and it should raise a red flag if you see one in P-chem.
 
  • #12
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.
 

What is the definition of logarithm of a number with units?

The logarithm of a number with units is the power to which a given base must be raised to equal that number with units. It is denoted as logb(x), where b is the base and x is the number with units.

Why do we use logarithms for numbers with units?

Logarithms are useful for numbers with units because they allow us to simplify complex calculations and compare numbers on a more manageable scale. They also help us to understand the rate of change of quantities with units.

How do logarithms with units relate to exponential functions?

The logarithm of a number with units is the inverse of an exponential function with the same base. This means that the logarithm of a number with units can be written as an exponent of the base, and vice versa.

What are some common applications of logarithms with units in science?

Logarithms with units are used in many scientific fields, including chemistry, physics, and biology. They are particularly useful for measuring quantities that vary exponentially, such as pH levels, sound intensity, and radioactive decay.

Can logarithms with units have negative values?

Yes, logarithms with units can have negative values. This occurs when the number with units is less than 1, and the base is greater than 1. In this case, the logarithm of the number with units is a negative number.

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