Why is the relation between coefficient of restitution and air pressure of a basketball logarithmic? Perhaps this is due to the spherical shape of the ball but yea, i've conducted the experiment for school and i'm trying to work through justifying this but need assistance. 
edit: here's the data

 A: The dependence of the coefficient of restitution (COR), $e$, on the gauge pressure of a ball, $P_G$, is modeled and experimentally verified in Can. J. Phys. 94: 42–46 (2016) (https://www.researchgate.net/publication/281791329_The_Coefficient_of_Restitution_of_Pressurized_Balls_A_Mechanistic_Model). The dependence the authors obtain is $$\frac{1+e^2}{(1-e^2)^2}=A P_G+B$$ (see the definitions of $A$ and $B$ in the article), so it is not exponential. However, an exponential function has the correct asymptotic behavior ($e\rightarrow 1$ when $P_G\rightarrow\infty$), so it can be a good fit.
A: Co-author of the article above here. Unfortunately
$$\frac{1+e^2}{(1−e^2)^2}=AP_G+B$$
isn't a very pretty function. However, concerning the 'trimmed sphere' approximation, it IS an approximation, but it is fairly representative of how balls actually deform on impact when you don't throw them too hard. See for example
https://www.youtube.com/watch?v=1HiXQrLigwo
The key points of the paper are there are two main restorative forces at play, the pressure force due to the compression of the ball due to the $PV = nRT$ gas law [by reducing the volume of the ball, you increase the pressure], and the forces due to the deformation of the walls of the ball [if you bend a flap of rubber, it wants to bend back]. For small deformations (meaning low impact velocities), both of these act like spring forces $F = -kx$ (equations 3 and 4 in the paper, $k$ is the part in front of $x$). And since the ball doesn't bounce back all the way up, there are some dissipative forces, which we model as a friction type of force. These dissipative forces are all the weird things a ball might do, but are mostly due to vibrations that dissipate into heat and sound.
For a high school experiment, if you want to connect this to our model, when you have $e$, for each data point, at each pressure, calculate $ \text{Big Fraction} = \frac{1+e^2}{(1−e^2)^2}$. Then if you do a plot of $\text{Big Fraction vs. Pressure}$, you should have a straight line (according to our model at least), with a slope of $A$ and a y-intercept of $B$.
You need a fairly precise way of measuring the coefficient of restitution $e$, otherwise the error bars on each point will be very big, and the graph won't look like anything useful.
