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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.

Ѕᴀᴀᴅ
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R. Emery
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  • Taking the base 2 log of both sides is the right approach. What was your answer? Why can't it be right? It is not a simple thing like plus or times – Ross Millikan Feb 12 '18 at 04:24

2 Answers2

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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.

Gaurang Tandon
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  • Okay I see what I did. The right hand side simply doesn't reduce. Still it seems like the mathematicians would create a function like that. Or at least give it a name. – R. Emery Feb 12 '18 at 04:29
  • @R.Emery "Still it seems like the mathematicians would create a function like that" You mean to say they would invent a new notation to depict such expressions? – Gaurang Tandon Feb 12 '18 at 04:31
  • I am saying that f(x,y) would be a very interesting and possibly a very useful function. – R. Emery Feb 12 '18 at 04:32
  • @R.Emery Well, given the way these notations work, if you ever write a book/research paper of your own, just stick this new $f(x,y)$ at the top of your book and then use it everywhere. If enough mathematicians find it useful, this new $f(x,y)$ might go viral in all maths books ;) – Gaurang Tandon Feb 12 '18 at 04:34
  • What do you mean by "taking the 2^x common"? – R. Emery Feb 12 '18 at 04:34
  • @R.Emery I meant $f(x,y)=\log_2((2^x)\cdot(1+2^{y-x}))$ which can be split as $\log(ab)=\log a+\log b$. – Gaurang Tandon Feb 12 '18 at 04:36
  • I wrote this:http://math.wikia.com/wiki/Generalization#Numbers or assembled it at least. I was thinking about putting it there. – R. Emery Feb 12 '18 at 04:36
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    @R.Emery I would not suggest putting a not-so-well known notation in a public wiki page. – Gaurang Tandon Feb 12 '18 at 04:37
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$$2^x + 2^y = 2^{f(x,y)},$$ $$2^x + 2^y = e^{f(x,y)\ln(2)},$$ $$\ln(2^x + 2^y) = f(x,y)\ln(2),$$ $$f(x,y)=\frac{\ln(2^x + 2^y)}{\ln(2)}$$

JJacquelin
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