Increased brightness vs. increased aperture / logarithmic eye

I understand that aperture is one of the most important specs on a
scope.
For a given scope design, is there a rule of thumb for the amount of
increased light gathering vs. increased aperture diameter (i.e. does
doubling the aperture result in doubling the amount of light
gathered)?

Also, I understand that the human eye perceives light logarithmically.
Is this true?

In the end I want to make a "perceived brightness vs. aperture
diameter" plot.

Thanks
Stephen