russian log problem

$\log _{ 5 }{ x+\log _{ \sqrt { 5 } }{ x } +\log _{ \frac { 1 }{ 5 } }{ x } =6 } \\ \log _{ 5 }{ x } +\frac { 1 }{ \frac { 1 }{ 2 } } \log _{ 5 }{ x } +\frac { 1 }{ -1 } \log _{ 5 }{ x } =6\\ \log _{ 5 }{ x } +2\log _{ 5 }{ x } -1\log _{ 5 }{ x } =6\\ \log _{ 5 }{ \frac { x\times { x }^{ 2 } }{ x } } =6\\ \log _{ 5 }{ { x }^{ 2 } } =6\\ x={ 5 }^{ 3 }\\ x=125$

Put all items in terms of base 5 logariths: $\log _{ \sqrt { 5 } }{ x } =\log _{ 5 }{ { x }^{ 2 } }$ and $\log _{ \frac { 1 }{ 5 } }{ x } =\log _{ 5 }{ { 5 }^{ -x } }$ . Then the left-hand side becomes ${ 5 }^{ x }{ x }^{ 2 }{ 5 }^{ -x }$ and the right ${ 5 }^{ 6 }$ . Cancelling and solving for $x$ we obtain $x=125$ .