Units in exponentials and logarithms

[mixj]

Hello, I was in class and came up with the question of: is there any physics formula in which a number with units is part of the exponent of said formula, and if there is how do the units behave?
Such as for example (x meters)^(y seconds)

Thank you in advance.

 

[Drakkith]

If I remember a thread on PF I perused through recently, things like exponentials and logs are essentially unitless. But I confess I don't remember for certain and don't have the link handy.

 

[Vanadium 50]

You can't. Think of the Taylor expansion of an exponential. If you are trying to take the exponential of something dimensionful, each term has a different dimension. Since that's nonsense, you can only take the exponentail of something dimensionless.

 

[Dale]

is there any physics formula in which a number with units is part of the exponent of said formula

No. The argument of an exponential function must be dimensionless. However, sometimes if the units are understood there can be an implicit division by the unit quantity. For example:

Such as for example (x meters)^(y seconds)

No, but you could have $$x^{\frac{y}{1\mathrm{\ s}}}$$ where $x$ is in meters and $y$ is in seconds.

 

[Baluncore]

Transcendental functions are often approximated by polynomials with integer exponents. Integration or differentiation of those polynomials conveniently maintains the integer exponents.

One common situation in physics that has a non-integer exponential is exponential decay, y = e^-t ; but then there is always a division by a time constant T, such as the half life, that eliminates the time unit from the exponent, y = e^-t/T .
That also holds for Fourier transforms.

 

[mixj]

Yes thank you, this question came about whilst looking at capacitor discharge and the time constants in which the seconds units also cancel.

That also holds for Fourier transforms

 

[DrClaude]

A good discussion of this can be found in

Chérif F. Matta, Lou Massa, Anna V. Gubskaya, and Eva Knoll
Can One Take the Logarithm or the Sine of a Dimensioned Quantity or a Unit? Dimensional Analysis Involving Transcendental Functions
J. Chem. Educ. 2011, 88, 1, 67–70
https://doi.org/10.1021/ed1000476

 

[vanhees71]

Take the definition of the exponential function in terms of its power series:
$$\exp x=\sum_{j=0}^{\infty} \frac{1}{j!} x^j,$$
you see that $x$ always must be a dimensionless quantity, because you cannot add quantities which have different units (in that case different powers of the unit of $x$). It just doesn't make any sense (neither mathematically nor physically).

The same holds true for the logarithm or trig functions, etc. Whenever you find a result, where they put a non-dimensionless quantity as the argument of such a function it's at least sloppy and most probably just wrong (or in Pauli's sense "not even wrong" ;-)).

 

[Baluncore]

A good discussion of this can be found in

Chérif F. Matta, Lou Massa, Anna V. Gubskaya, and Eva Knoll
Can One Take the Logarithm or the Sine of a Dimensioned Quantity or a Unit? Dimensional Analysis Involving Transcendental Functions
J. Chem. Educ. 2011, 88, 1, 67–70
https://doi.org/10.1021/ed1000476

Unfortunately, that paper has no relevance or credibility as a publication, because it is behind a paywall. If it was worth reading, it would be an open publication. The abstract contains no substance, reading more like a statement of intent, or maybe just clickbait.

 

[Dale]

that paper has no relevance or credibility as a publication, because it is behind a paywall. If it was worth reading, it would be an open publication

That is not a generally true statement. In fact, as a broad rule of thumb I would say the opposite is more typical in my experience.

 

[Vanadium 50]

You can't. Think of the Taylor expansion

Take the definition of the exponential function in terms of its power series:

Couldn't have said it better myself!