Trying to solve $\log_2(x-1)=\log_3(x+1)$ and can't seem to get it algebraically. Tried changing bases, moving things around, but can't seem to crack it.
Can all logarithm problems be solved algebraically?
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logarithms
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0To answer the question in the title: nope. Even things as simple as $x\log\,x=3$ require the use of a different sort of function for expressing solutions. For your actual problem: barring a fair bit of cleverness, I see no straightforward solution... – 2012-02-10
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0@J.M. are there any results which show that? Something similar to [Abel–Ruffini theorem](http://en.wikipedia.org/wiki/Abel%E2%80%93Ruffini_theorem) regarding algebraic solution of polynomial equation? – 2012-02-10
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0The keyword here is [transcendental equations](http://en.wikipedia.org/wiki/Transcendental_equation). AFAIK there is no general algorithm to solve them. Only tricks to solve specific ones.. I can't seem to find any theorems on that though. – 2012-02-10
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0@J.D. Not sure about general, but the example I gave requires the Lambert function to express the solution. Only slightly more complicated transcendental equations don't even have the luxury of an "easy" closed form... – 2012-02-10