# In the one-variable case we can take a log transform of the function f ( x ) &#x2192;

In the one-variable case we can take a $\mathrm{log}$ transform of the function $f\left(x\right)\to \mathrm{log}\left(f\left(x\right)\right)$ and know that the same value maximizes both because log is increasing. The log turns products into sums which facilitates taking derivatives. Can the same argument be applied for multivariable cases? The question is: for a function $f\left(x,y\right)$ if I wish to find the ${x}^{\star },{y}^{\star }$ that maximize $f\left(x,y\right)$ are these the same ${x}^{\star \star },{y}^{\star \star }$ that maximize $\mathrm{log}\left(f\left(x,y\right)\right)$? My intuition is yes, nothing should be different in this multivariable case, but I desire a quick sanity check before setting off with computations. I would appreciate a rigorous proof or a counterexample.
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heilaritikermx
Yes (as long as $f$ takes positive values) since $f\left({x}^{\ast },{y}^{\ast }\right)\ge f\left(x,y\right)$ if and only if $\mathrm{log}f\left({x}^{\ast },{y}^{\ast }\right)\ge \mathrm{log}f\left(x,y\right)$.