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Find dimensional formula of $\alpha$ [duplicate]

The power delivered by a force is given by the relation $$P=\frac{\alpha}{\beta}e^{-\beta t},$$ where $t$ is time. Find the dimensional formula for $\alpha$. So, $-\beta t$ doesn't make sense for the ...
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Is it a problem that you can write the logarithm of a quantity with units? [duplicate]

While working out something in thermodynamics, I encountered an equation that had a term like $\log(n_1/n_2)$, where, $n_1$ and $n_2$ are the number densities. Now of course the argument of the $\log$ ...
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Dimensional analysis - application to logarithms

I read some nice threads about this topic: physics StackExchange maths StackExchange stats StackExchange However, it still puzzles me that logarithm of some physical quantity has no units. Example, ...
How do I decide the dimensions of a trigonometric quantity and a logarithmic quantity? For example, what are the dimensions for: $$\frac{C}{B} = \frac{D^2}{A} + \log \left(\frac{AC}{BD}\right)$$