How to find full width at half max for non-Gaussian peaks.

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I have a signal that has several distinct peaks. I need to characterize these peaks by their max y value, their x location, and their width. I thought FWHM would be a good measure. However, these peaks are just from a data logger. They are pretty Gaussian curves. Furthermore, the curve doesn't go to zero in between peaks. The peaks are more of like distinct bumps on a 1/ln(x) curve. Any idea on how characterize the width of these peaks?

Thanks
 
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If you don't have a function that you can fit the background width (which would be better), and if the peaks all have qualitatively the same shape, then you could try calculating the curvature (second derivative) at the tip of the peak.