Semi-elasticities in Regressions

Suppose you have a log-linear regression like this:

log(y) = beta*x

The way to interpret beta is as the percentage change in y that we get from a 1 unit change in x. To see that note that the regression equatino is the same as y = exp(beta*x), in which case dy/dx = beta*exp(beta*x). Thus, the percentage change in y when x changes is (dy/dx)/y = (beta*exp(beta*x))/exp(beta*x) = beta.

This contrasts with the log-log form, log(y) = beta*log(x), in which case beta is the elasticity of y with respect to x, i.e., the percentage change in y that we get from a 1 percent change in x.