if you look at the median, interquartile range and mean of your data that should let you decide how it is distributed.

The problem is that there is no expectation that the input values will or should be evenly distributed. Ie. They could be clumped at both ends, or predominantly more at one end than the other; or any other variation.

My gut tells me you're pointing me in the right direction; but I'm not seeing how to make use of the suggestion?

I'm not sure how you could detect the base of log data, it will all look the same as it's trivial to convert from one base to another.

Indeed. The log base doesn't matter. The point was only that certain datasets are very obviously logarithmic by (human) inspection.