Regression of first differenced log-transformed model
What you describe is very standard in academic econometric practice. It essentially answers what is the elasticity of Y with respect to X identified from changes in X and Y. 

Here are a few problems that can be introduced by this process


First-differencing is very similar to $Y_t = \alpha + \beta_0 * Y_{t-1} + \beta_1 * X_{t} + \beta_2 * X_{t-1} + \epsilon$ with the additional restrictions that $\beta_0 = 1$ and $\beta_1 = -\beta_2$
First-differencing can introduce serial correlation in standard errors