How are stellar elemental abundances quoted? Close to the bottom of page 4 of this article (marked as page 164 in the upper left corner) states 


  Values are given in the logarithmic scale usually adopted by astronomers, A$_{e\ell}$ = log N$_{e\ell}$/N$_H$ + 12.0, where N$_{e\ell}$ is the abundance by number.


I have never used stellar elemental abundances in this way, so I want to check a few things.

First of all, is the logarithm used by this paper log$_{10}$ or log$_e$ (often written ln)?  My guess is log$_e$ because of what follows.

Let's say I want to calculate the elemental abundance of He relative to H.  I believe the correct math would be to do the following

$$\frac{\textrm{N}_{He}}{\textrm{N}_H} = \textrm{exp}(\textrm{A}_{He} - 12.0)$$

Is this the correct way to calculate the relative abundances?  This equation gives a value of 0.343 for the helium abundance relative to hydrogen, which seems to be the correct value for the sun (the sun is usually quotes as being about 70% H and about 25% He, which gives a relative abundance of 0.357).  This correct value is what leads me to believe that the log is log$_e$.
First of all, your first question.

This source clearly state that Values are given in the usual logarithmic (dex) scale, for the same formula that you quoted (similar job).
It is a bit tricky as the article "explains" the values, but you have to pay attention to the exact definition.

I think it is better to work out with an example. Let's take the He. 

Good enough, you can better read the paper from here.

From the table, we have $A_{el}=10.93$. This is the abundance of He relative to H (in logarithmic scale).
From this you find out that $\frac{N_{el}}{N_H}=0.08 = 8\%$.
Indeed the work confirms this value (see the last page).

What you quote as about 25% He, is what they call abundances by mass of [...] Helium (Y), which means $Y = mass\ of\ Helium / mass\ of\ Hydrogen$, and this is indeed about $25\%$.