# Log-Linear Model

I know that the log-linear model shows the percent change in y if there is a one-unit change in x but how would you solve for it the other way to show a percent change in x if there is a one-unit change in y?

In principle you can just take the model coefficients and solve for $$\ln x$$. If you have:
$$\ln y = \hat{\beta}_0 + \hat{\beta}_1 \ln x \implies \ln x = -\frac{\hat{\beta}_0}{ \hat{\beta}_1} + \frac{1}{ \hat{\beta}_1} \ln y$$
However, this being said even though you can do this mathematically the OLS assumes that the relationship is exogenous, so solving for $$\ln y$$ would not allow you to interpret the $$1/ \hat{\beta}_1$$ as a causal effect - it is just inverse of the effect of $$\ln x$$.
If you believe that there is relationship that goes both ways you should be using some model that allows for that. A vector autoregression (VAR) would be one example of such model. VAR would estimate the relationship going both ways giving you separate set of coefficients when $$\ln x$$ is the dependent variable. However, this is just one example, there is a cornucopia of models that can help deal with endogeneity. For overview of them you can see Verbeek (2008) A Guide to Modern Econometrics, 4th ed.