$$ \begin{aligned} & P=\frac{\alpha}{\beta} \log _e\left(\frac{k t}{\beta x}\right) \\ & \frac{k t}{\beta x}=1 \Rightarrow \beta=\frac{k t}{x}=\frac{M L^2 T^{-2}}{L} \\ & \left(\because E=\frac{1}{2} k t\right) \end{aligned} $$ As P is dimensionless $$ \Rightarrow[\alpha]=[\beta]=\left[M L T^{-2}\right] $$