5^(1+log_4x)+5^{log_(0.25)x-1}=(26)/(5)

Question

`5^(1+log_4x)+5^{log_(0.25)x-1}=(26)/(5)`

Answer 1

`\displaystyle 5^(1+\log_4 x)+5^(-\log_4x-1) = (26)/(5)\\ \\ \text{Sea } z=5^(1+\log_4 x) \text{ entonces}: \\ \\ z+(1)/(z)=(26)/(5)\\ \\ 5z^2-26z+5=0 \\ \\ (5z-1)(z-5)=0 \\ \\ z\in\left\{(1)/(5);5\right\}\\ \\ \text{Then: }\\ \\ 5^(1+\log_4x)=5^(-1) \vee 5^(1+\log_4x)=5\\ \\ 1+\log_4x=-1 \vee 1+\log_4x=1\\ \\ \log_4x =-2 \vee \log_4x=0 \\ \\ x=(1)/(16) \vee x=1\\ \\ \text{Answer: }x\in\left\{(1)/(16);1\right\}`