This is the exponential function, 2^x. Where the independent variable is the number of walls jumped. For instance, after 30 jumps, your score/tile reads 1073741824, or 2^30.
"Therefore, despite these two functions being the inversely related, the correct relationship is 2^x."
There is no "correct relationship", there is a relationship that (depending on choice of variables) we can describe as y=2^x or y=lg(x).
However, as you said, one choice of describing the relationship is certainly more natural. The exponential description assigns a score to every natural number. While the logarithmic description does not.
I would interpret correct in the way that it is sometimes said, "the correct way to think about X is..." which doesn't say other ways of thinking are wrong, but limited or not illuminating.