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Question Number 170054 by daus last updated on 15/May/22

Answered by greougoury555 last updated on 15/May/22

$$\{\begin{array}{l}a=9^x\\ b={12}^x\\ a+b={16}^x\end{array}\Rightarrow 9^x+{12}^x={16}^x$$$$\Rightarrow {\left(\frac{9}{12}\right)}^x+1={\left(\frac{16}{12}\right)}^x$$$$\Rightarrow {\left(\frac{3}{4}\right)}^x+1={\left(\frac{4}{3}\right)}^x;{\left(\frac{4}{3}\right)}^x=\left(\frac{b}{a}\right)=y$$$$\Rightarrow \frac{1}{y}+1=y;y^2-y-1=0$$$$\Rightarrow y=\frac{1+\sqrt{5}}{2}=\frac{b}{a}$$

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