Log transformation in constrained optimization

Question

In constrained maximization, should I log-transform the constraint if the objective function is log-transformed?

$$\max_{x, \ y} \,\, x^\alpha y^{1 - \alpha} \qquad \text{s.t.} \qquad p_x x + p_y y \ \leq \ p_x \hat{x} + p_y \hat{y} $$

Here if I log-transform $x^\alpha y^{1 - \alpha} $, then how should I deal with the constraint?

Thank you for the help!

Answer 1

You can just take the log of the objective, assuming it is positive everywhere, and not change the constraints. Any monotonic transformation of the objective is fine since it doesn't change the maximum.