Are logarithms the only function for which f(xy)=f(x)+f(y) is true?
(Mathematical foundations of thermodynamics)

Question

Are logarithms the only function for which f(xy)=f(x)+f(y) is true?
(Mathematical foundations of thermodynamics)

frostbite · Answer

In addition we require the function to be continuous and non-discrete.

frostbite · Answer

@Michele_Laino

michele_laino · Answer

from such identity, comes the additivity of the entropy

frostbite · Answer

So far I have figured out that yes. I was simply wondering if there were other functions in an attempt to understand why it ended being the natural logarithm that was used. @Michele_Laino

rational · Answer

The second thing that comes to my mind after logs is the argument of a complex number :

$$\arg(z_1z_2) = \arg(z_1)+\arg(z_2)$$

anonymous · Answer

Also, not too helpful, but the trivial constant functions $f(x)=0$ and $f(x)=2$ work too (though is the second one actually trivial?).

kainui · Answer

There's a number theory function that obeys this relationship but for positive integers only, 

$$\Omega(a*b)=\Omega(a)+\Omega(b)$$
$\Omega(n)$ is the number of prime divisors in n, for instance, 

$\Omega(12)=\Omega(2*2*3) = 3$

since it has 3 prime divisors, or we can expand it like this:

$\Omega(12)=\Omega(2*2*3)=\Omega(2)+\Omega(2)+\Omega(3) = 1+1+1$

Although it's not so much different than the logarithm function, really.

thomas5267 · Answer

https://en.wikipedia.org/wiki/Logarithm#Logarithmic_function More precisely, the logarithm to any base b > 1 is the only increasing function f from the positive reals to the reals satisfying f(b) = 1 and [31]$$ f(xy)=f(x)+f(y)$$