solving equations with powers

Question

Im trying to solve the equation $$3\cdot2^{-2/x} + 2\cdot9 ^{-1/x} = 5\cdot6^{-1/x }$$ So far I tried applying logaritmas but it didnt prove helpful...are there any other ways?

Answer 1

Since $9=3^2$ the equation becomes

$$3\times 2^{-2/x}+2\times 3^{-2/x}=5\times2^{-1/x}3^{-1/x}$$ so let $a=2^{-1/x}$ and $b=3^{-1/x}$ then we have

$$3a(a-b)+2b(b-a)=0\iff(a-b)(3a-2b)=0$$ and since $a\ne b$ then $3a-2b=0$ hence $$\frac a2=\frac b3\iff2^{-1/x-1}=3^{-1/x-1}\iff\left(\frac1x+1\right)(\ln3-\ln2)=0\iff x=-1$$

Answer 2

Set $2^{-\frac{1}{x}} = a, \ 3^{-\frac{1}{x}} = b$ so the equation becomes $3 a^2 + 2 b^2 -5 ab = 0$.