Not exponentially, just inverse-quadratically :)

Presumably something that got exponentially worse with proximity would have badness e^(-kr) where r is distance and k is some constant. So depending on k, it’s worse by some constant amount when you bring d down from 1 meter to zero. This is, notably, finitely worse.

But k/r^2 (different k) is a whole different beast. It’s infinitely worse at zero distance!

Of course, the radioactive source itself is not a point, and the human body isn’t a point either, so it’s not infinitely bad at zero range. Closer than a meter or so (very roughly the size of a body), it will merely concentrate the exposure over a smaller portion of the target and deliver a larger fraction of its total output to the target. The latter effect is a constant factor not vastly greater than 1 when comparing 1 meter to zero meters.

> I was thinking, 10 x-rays per hour doesn’t sound so terrible.

I believe the popular term is "not great, not terrible" ;)