In many resources/videos I see comments being made along the lines of "and we can see here that we have a logarithm/exponential so this will be an expensive computation to make." (such as with the sigmoid activation function for a NN). What is behind these statements?

A more specific example would be: what makes $e^{-x}$ so much different than something like $x^4$ or $4^x$ in a computer calculation?