<s>[INST] Where am I going wrong while trying to solve this logarithmic equation? $$\log _{ 0.2 }{ x } +\log _{ \sqrt { 5 }  }{ x } =\log _{ 25 }{ x } +1$$
Steps I took:
$$\log _{ \frac { 1 }{ 5 }  }{ x } +\frac { \log _{ \frac { 1 }{ 5 }  }{ x }  }{ \log _{ \frac { 1 }{ 5 }  }{ \sqrt { 5 }  }  } =\frac { \log _{ \frac { 1 }{ 5 }  }{ x }  }{ \log _{ \frac { 1 }{ 5 }  }{ 25 }  } +1$$
$$\log _{ \frac { 1 }{ 5 }  }{ x } +-\frac { \log _{ \frac { 1 }{ 5 }  }{ x }  }{ 2 } =-\frac { \log _{ \frac { 1 }{ 5 }  }{ x }  }{ 2 } +1$$
I can keep going, but this doesn't seem to lead me to the correct answer. Where did I go wrong? 
 [/INST] at line 3: $\log _{ \frac { 1 }{ 5 }  }{ \sqrt { 5 }  }$ doesn't equal -2, but $-\frac {1}{2}$, so the third line need to be:
$\log _{ \frac { 1 }{ 5 }  }{ x } +-2\log _{ \frac { 1 }{ 5 }  }{ x } =-\frac { \log _{ \frac { 1 }{ 5 }  }{ x }  }{ 2 } +1$
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