I am trying to reproduce the mass function graph from a paper. I have calculated the mass function $\frac{\rm{d}f}{\rm{d}\log M}$, where $f$ is the mass fraction with respect to the total mass in the universe and $M$ is the mass of an astrophysical object I am interested in. I am plotting $\frac{\rm{d}f}{\rm{d}\log M}$ with respect to $\log M [M_\odot]$. The paper seems to show that the units of $\frac{\rm{d}f}{\rm{d}\log M}$ are [1/dex]. How do I interpret [1/dex] and how do I convert my raw values of $\frac{\rm{d}f}{\rm{d}\log M}$ that I computed into these [1/dex] units?
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$\begingroup$ It is just in there to tell you that by log the mean log to the base ten rather than the natural logarithm. One can also infer that $f$ is unitless as $d/d\log(M)=\ln(10)M d/d M$. $\endgroup$– VirgoCommented Dec 18, 2023 at 2:27
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Dex is short for 'decimal exponent'. [1/dex] is a unit often used in the context of functions with respect to the logarithm of a quanity in base 10.
If a function has a value of $Y$ [1/dex] at a certain $ log M $, it means that for a factor of 10 change in $ M $, the function changes by $ Y $.
For instance, if $ \frac{df}{dlogM} $ = 0.5 [1/dex] at $ log M = 1 $ (where $ M $ is in solar masses), it indicates that the mass fraction $ f $ increases by 0.5 when $ M $ increases by a factor of 10.
Your raw data is likely already in units of [1/dex] if the logarithm used was base 10.
