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Why is pseudorapidity defined as $\eta = -\ln[\tan(\frac{\theta}{2})]$ as opposed to just $\eta = \tan(\frac{\theta}{2})$?

user13948
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Taking the concept of rapidity first, which brings in the Ln term. The same argument can be made, with more convoluted math, for pseudorapdity

Define $y = 1/2 \ln \frac{E+p_zc}{E-p_zc}$

In collisions, the energies involved are obviously very high. If we make the assumption that a particular particle is made to travel along the XY plane that is perpendicular to the direction of propagation of the beam then, in this case, this will result in a low value for $p_z$, with a rapidity close to 0, that is $\ln 1$.

If we instead direct the same highly relativistic part down the axis of the beam, in the usual +z direction, then we will have a situation where the same highly relativistic particle travels predominantly down the beam axis, say in the +z direction. In this case, $E= p_zc$, and $y → +∞$.

The upshot of all this is that the rapidity tends to zero when a particle is approximately transverse to the beam axis, but tends to $±∞$ when a particle is moving close to the beam axis in either direction.