Are there practical uses (ie: in engineering, applied sciences, chemistry, IT, etc) for using non-integer bases?
From other questions on the topic, I see that it's just another way of representing numbers, but does this ever come up in practical examples? I know that taking the base-2 logarithm of a number lets us calculate the number of entropy bits in a password, the base-10 logarithm has uses in power calculations, and the natural logarithm can be used for things like determining RC filter decay time constants.
So, with the exception of the natural logarithm, does it ever make sense to use non-integer bases?
Well the function $e^x$ is extremely important and basic in all of science and so therefore so is the natural logarithm (which is log base $e$). And of course $e$ is not an integer.