# Separating $x_1^{x_2}$ into sum of two terms by variable transformation

Given the function $f(x_1, x_2)$ in the form of product of two variables.

$$f(x_1, x_2) = x_1^{x_2}$$

I want to apply a variable transformation on this function, so that I can write it sum or difference of two terms.

How can I do it?

Example:

Given

$$f(x_1, x_2) = x_1x_2.$$

We define:

$$y_1(x_1,x_2)=\frac{1}{2}(x_1+x_2) \\ y_2(x_1,x_2)=\frac{1}{2}(x_1-x_2)$$

Now we can write $f(x_1,x_2)$ as a sum of two different terms like below:

$$f(x_1,x_2) = g\Big(y_1(x_1,x_2),y_2(x_1,x_2)\Big) = y_1^2 - y_2^2$$

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## 1 Answer

Take the $\log$ of both sides. Then the transformation is the same as you defined. However at the end you need to include exponential.

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