<s>[INST] exponential equation: $6^x+8^x+15^x=9^x+10^x+12^x$ What are the solutions of this equation? Or at least in which interval are they? $$6^x+8^x+15^x=9^x+10^x+12^x$$ I tried to find an increasing function, or use some inequalities but I got nothing out of it...
 [/INST] Let $a=3^x$ and $b=2^x$ and $c=5^x$.
Then we have that
$$ab+b^3+ac=a^2+bc+ab^2$$
$$ab+b^3+ac-a^2-bc-ab^2=0$$
$$a(b-a)+b^2(b-a)-c(b-a)=0$$
$$(b-a)(a+b^2-c)=0$$
Now $a=b \implies x=0$ and $a+b^2-c=0 \implies  3^x+4^x=5^x$ which gives solution for $x=2$ only and not for any higher integer $x$ by Fermat's Last Theorem.
So these are the $2$ solutions.
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