1. We are given $P=\frac{\alpha}{\beta}\ln\left(\frac{kt}{\beta x}\right)$ and $P$ is dimensionless.

2. Since the logarithm function can take only a dimensionless argument, $\frac{kt}{\beta x}$ must be dimensionless.

So, $[kt]=[\beta x]$

3. Now find dimensions of each quantity:

• Boltzmann constant $k$ has dimensions of energy per temperature: $[k]=\frac{[\text{energy}]}{[\text{temperature}]}=\frac{[ML^2T^{-2}]}{[\Theta]}$
• Temperature $t$ has dimension $[\Theta]$

Hence, $[kt]=[ML^2T^{-2}]$

Also, $x$ is distance, so $[x]=[L]$

Therefore, $[\beta][L]=[ML^2T^{-2}]$ $[\beta]=[MLT^{-2}]$

4. Since $P$ is dimensionless and $\ln(\cdot)$ is already dimensionless, the factor $\frac{\alpha}{\beta}$ must also be dimensionless.

Thus, $[\alpha]=[\beta]=[MLT^{-2}]$

5. Compare with options:

• A: $[M^0L^{-1}T^0]$
• B: $[ML^0T^{-2}]$
• C: $[MLT^{-2}]$
• D: $[ML^2T^{-2}]$

So the correct option is: $\boxed{C}$