Log-Linear Model

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.