An expression for a dimensionless quantity P is given by P = α β log e k T β x ; where α…

Given $p=\frac{\alpha }{\beta }\ln \left(\frac{kT}{\beta x}\right)$

We know that expression inside log is dimensionless.

Therefore,$\left[\frac{kT}{\beta x}\right]$ is dimensionless

$\Rightarrow \left[\beta \right]=\left[\frac{kT}{x}\right]={\mathrm{ML}}^2{\mathrm{T}}^{-2}{\mathrm{K}}^{-1}\times \mathrm{K}\times {\mathrm{L}}^{-1}={\mathrm{MLT}}^{-2}$

Since $p$ is dimensionless $\therefore \left[p\right]=\left[\frac{\alpha }{\beta }\right]={\mathrm{M}}^0{\mathrm{L}}^0{\mathrm{T}}^0$  $\Rightarrow \left[\alpha \right]=\left[\beta \right]$   $\Rightarrow \left[\alpha \right]={\mathrm{MLT}}^{-2}$