What should I do? If we divide say with $3^{x-1}$, we win nothing, considering the $34$. How do we solve this equation?
Thanks.
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8The function is monotonous. Since $x=3$ is a solution, it is the only one. – Dietrich Burde Jan 02 '16 at 17:27
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Note that the function on the left is increasing and there is an obvious solution to $3^y+5^y=34$ – Mark Bennet Jan 02 '16 at 17:28
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1@DietrichBurde: Is that also a way of saying monotonic? Just curious. – Brian Tung Jan 02 '16 at 18:21
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@BrianTung Google suggest "continuous function" when asking "monotonous function". So there is only "monotonic function", I fear. – Dietrich Burde Jan 02 '16 at 18:43
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It is clear at first sight that $x=3$ is a solution so the only one by the increasing function $3^x+5^x$. Trying to find a "deduction" we see first at the integers and from $$3^x+(3+2)^x=3^x+3^x +K$$ where K is positive, one has $2<x<4$ because $3^4+3^4>34$ hence the end (although it had been more interesting that x be an irrational).
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Hint: $3^{x-1}+5^{x-1}=3^2+5^2 $ also, think about the graph of $\alpha ^x$, what you can dedude?
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