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 09 '12 at 04:00
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$x,y$ are what? Natural numbers? – GEdgar Mar 09 '12 at 04:07
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any positive real numbers – Keneth Adrian Mar 09 '12 at 04:12
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There is no expression for $x$ in closed form as an elementary function of $y$.
Gerry Myerson
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3How come? is there a proof why it has no closed form? If not elementary, is there a closed form? thanks – Keneth Adrian Mar 09 '12 at 04:41
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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 09 '12 at 08:57
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thanks. But, is there a form extending from elementary functions? – Keneth Adrian Mar 09 '12 at 09:05
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Not that I know of. But I could be wrong - not every function is my personal friend. – Gerry Myerson Mar 09 '12 at 12:27