When would it be appropriate to perform a non-linear transformation on your data?

There seem to me to be two main reasons to try a non-linear transformation on your data:

(1) The data itself is obviously non-linear. e.g. When plotted on a linear scale, the points follow a non-linear curve.

(2) The data pertains to a non-linear system. e.g. population growth.

If you suspect an exponential relationship like `y = a*b^x` then try linear regression on `x` vs `log(y)`.

If you suspect a power relationship like `y = ax^b` then try linear regression on `log(x)` vs `log(y)`.

If you suspect a polynomial relationship like `y = ax^2+bx+c` and you have data points at regular `x` intervals (e.g. periodic samples), then you can use the differences between successive pairs of samples as a new sample to reduce the degree by `1`. Repeat as necessary until the resulting data looks linear.