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Is there a way to solve for $x$ given the function: $$y= 3^x-2^x$$ in terms of $y$?

I tried a lot of algebraic manipulations but I ended up nothing. Or, should we say it it impossible to do so?

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It's $\rm\ Z^C - Z\: =\: Y\ $ for $\rm\ Z = 2^X,\:\ C = log_2 3$. –  Bill Dubuque Mar 9 '12 at 4:00
    
$x,y$ are what? Natural numbers? –  GEdgar Mar 9 '12 at 4:07
    
any positive real numbers –  Keneth Adrian Mar 9 '12 at 4:12
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1 Answer 1

up vote 2 down vote accepted

There is no expression for $x$ in closed form as an elementary function of $y$.

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How come? is there a proof why it has no closed form? If not elementary, is there a closed form? thanks –  Keneth Adrian Mar 9 '12 at 4:41
    
These proofs aren't easy, and I certainly can't come up with one myself. Do a search for terms like "closed form" and "elementary function" and you may see what you are up against. –  Gerry Myerson Mar 9 '12 at 8:57
    
thanks. But, is there a form extending from elementary functions? –  Keneth Adrian Mar 9 '12 at 9:05
    
Not that I know of. But I could be wrong - not every function is my personal friend. –  Gerry Myerson Mar 9 '12 at 12:27
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