Hi! I'm having trouble understanding why you interpret the "log" part in the log-lin, lin-log, and log-log models as a percentage change. Let's say you have the regression equation Y = Beta0 + Beta1log(x). I read that if you take the derivative of Y with respect to x you get Y' = Beta1 x (deltax/x), so the change in Y is equal to the coefficient estimate times the percentage change in x. What does the derivative of Y have anything to do with what value I plug in for x to get Y? I know that in a log-lin model as x increases, Y grows at a decreasing rate. But I don't how that is relevant to the regression equation.
So basically I'm asking why, if you have the regression equation Y = 30 + 50log(x), do you interpret the second term on the right-hand side as a percentage change in x is on average associated with a 0.5 unit increase in Y. I know that part of the derivative (deltax/x) mentioned above has everything to do with it, but I can't make the connection. Why am I looking at the rate of change? Can someone please explain it to me as simply as possible? Thanks.