[tex]16^x+20^x=25^x|\div 16^x\\\\1+\dfrac{20^x}{16^x}=\dfrac{25^x}{16^x}\\\\1+\left(\dfrac{20}{16}\right)^x=\left(\dfrac{25}{16}\right)^x\\\\1+\left(\dfrac{5}{4}\right)^x=\left(\dfrac{25}{16}\right)^x\\\\\left(\dfrac{25}{16}\right)^x-\left(\dfrac{5}{4}\right)^x-1=0\\\\\left(\left(\dfrac{5}{4}\right)^x\right)^2-\left(\dfrac{5}{4}\right)^x-1=0\\\\\\t=\left(\dfrac{5}{4}\right)^x\\\\t^2-t-1=0[/tex]
[tex]\Delta=(-1)^2-4\cdot1\cdot(-1)=1+4=5\\\sqrt{\Delta}=\sqrt5 \\t_1=\dfrac{-(-1)-\sqrt{5}}{2\cdot1}=\dfrac{1-\sqrt5}{2}\\t_2=\dfrac{-(-1)+\sqrt{5}}{2\cdot1}=\dfrac{1+\sqrt5}{2}\\\\\\\left(\dfrac{5}{4}\right)^x=\dfrac{1-\sqrt5}{2}\\x\in\emptyset\\\\\\\left(\dfrac{5}{4}\right)^x=\dfrac{1+\sqrt5}{2}\\\\x=\log_{5/4}\dfrac{1+\sqrt5}{2}[/tex]
