Logarithm and sum of powers

Question

If $$2^x + 2^y = 2^{f(x,y)},$$ then $f(x,y) = $?

I tried to take logarithm of both sides but ended up with an answer that I know cant be right.

Answer 1

You must take the $\log_2$ of both sides, since that's the only way you can get the $f(x,y)$ in polynomial/solve-able form. You'll get:

$$f(x,y)=\log_2(2^x+2^y)$$

And that's pretty much it. We can't apply any logarithm property here. This is one such expression where taking the $2^x$ common may result in lesser readability than it currently has, hence, leaving it in this form is much better.

Answer 2

$$2^x + 2^y = 2^{f(x,y)},$$ $$2^x + 2^y = e^{f(x,y)\ln(2)},$$ $$\ln(2^x + 2^y) = f(x,y)\ln(2),$$ $$f(x,y)=\frac{\ln(2^x + 2^y)}{\ln(2)}$$