Applying log transform to find critical points when there is a constraint

Question

I know that the critical points of a function $f$ can be found by differentiating either $f$ or $\ln f$, as for example when maximizing likelihood. But what about when a constraint is added. For eg.:

$$\min_\mathbf{x}f \:\:\:\:s.t. \sum_i x_i=C$$

where $\mathbf{x}$ is a vector of arguments. Will differentiation of the following two lagrangians give the same minimum?

$$\mathcal{L}=f+\lambda (\sum_i x_i-C)$$

$$\mathcal{L}=\ln f+\lambda (\sum_i x_i-C)$$

Answer 1

Any strictly monotonic transformation of the objective function does not affect the location of the optimum. Hence, yes, both lagrangians will give the same minimizer.