Correct option is (B) 1/5, 25
\(x^{1-log _5\,^x}\,=0.04\) \(=\frac4{100}\) \(=\frac1{25}=5^{-2}\)
\(\Rightarrow\) \(log_5\,(x^{1-log _5\,^x})=log_5\,5^{-2}\) (By taking \(log_5\) both sides)
\(\Rightarrow\) \((1-log _5\,^x)\,log_5\,^x=-2\,log_5\,5\) \((\because log\,a^n=n\,log\,a)\)
\(\Rightarrow\) \(log _5\,^x-(log_5\,^x)^2=-2\) \((\because log_5\,5=1)\)
\(\Rightarrow\) \((log_5\,^x)^2-log _5\,^x-2=0\)
\(\Rightarrow\) \((log_5\,^x)^2-2\,log _5\,^x+log _5\,^x-2=0\)
\(\Rightarrow\) \((log_5\,^x)\,(log_5\,^x-2)+(log_5\,^x-2)=0\)
\(\Rightarrow\) \((log_5\,^x-2)\,(log_5\,^x+1)=0\)
\(\Rightarrow\) \(log_5\,^x-2=0\) or \(log_5\,^x+1=0\)
\(\Rightarrow\) \(log_5\,^x=2\,or\,log_5\,^x=-1\)
\(\Rightarrow\) \(x=5^2=25\) or \(x=(5)^{-1}=\frac15\)
