# Intuition behind the change of variable of BS into Heat Equation

$$\tau=T-t$$ $$u=Ce^{r\tau}$$ $$x=\ln(\frac{S}{K})+(r-\frac{1}{\sigma^2})\tau$$ The first transform is reverse in time. The second is reversely discounted call price. But I find it hard to see the intuition behind the third transformation. And I am also wondering why this is not a one-to-one transform of time, call price, and price of the underlying. Why is 'multiplying by a Jacobian' doesn't work in this case?

$$y = \log (S/K)$$ $$x= y + (r-0.5\sigma^2)\tau$$
The second one is remove the drift. The log has drift $r-0.5\sigma^2$ so removing that changes to coords in which the log stock is driftless.