$\begin{align}  & 6({{9}^{\tfrac{1}{x}}})-13({{6}^{\tfrac{1}{x}}})+5({{4}^{\tfrac{1}{x}}})=0 \\ & 6{{({{3}^{\tfrac{1}{x}}})}^{2}}-13({{2}^{\tfrac{1}{x}}}\times {{3}^{\tfrac{1}{x}}})+5{{({{2}^{\tfrac{1}{x}}})}^{2}}=0 \\ & \text{Let }{{\text{3}}^{\tfrac{1}{x}}}=a,\text{ }{{2}^{\tfrac{1}{x}}}=b \\ & 6{{a}^{2}}-13ab+5{{b}^{2}}=0 \\ & 6{{a}^{2}}-10ab-3ab+5{{b}^{2}}=0 \\ & 2a(3a-5b)-b(3a-5b)=0 \\ & (2a-b)(3a-5b)=0 \\ & 2a=b\text{ or 3}a=5b \\ & \frac{b}{a}=2\text{ or }\frac{b}{a}=\frac{3}{5} \\ & \frac{{{3}^{\tfrac{1}{x}}}}{{{2}^{\tfrac{1}{x}}}}=2 \\ & \text{Take log of sides} \\ & \text{log}\frac{{{3}^{\tfrac{1}{x}}}}{{{2}^{\tfrac{1}{x}}}}=\log 2 \\ & \log {{\left( \frac{3}{2} \right)}^{\tfrac{1}{x}}}=\log 2 \\ & \frac{1}{x}\log 1.5=\log 2 \\ & \frac{1}{x}=\frac{\log 2}{\log 1.5} \\ & x=\frac{\log 1.5}{\log 2}\text{ or} \\ & \frac{b}{a}=\frac{3}{5} \\ & \frac{{{2}^{\tfrac{1}{x}}}}{{{3}^{\tfrac{1}{x}}}}=\frac{3}{5} \\ & {{\left( \frac{2}{3} \right)}^{\tfrac{1}{x}}}=\frac{3}{5} \\ & \text{Take log of both side} \\ & \text{log}{{\left( \frac{2}{3} \right)}^{\tfrac{1}{x}}}=\log 0.6 \\ & \frac{1}{x}\log \frac{2}{3}=\log 0.6 \\ & \frac{1}{x}\log (0.6667) \\ & \frac{1}{x}=\frac{\log 0.6}{\log 0.6667} \\ & x=\frac{\log 0.6667}{\log 0.6} \\\end{align}$