New Logarithmic Proof – Quicker Version

Prove that:

\log _{ a }{ x } =\frac { \log _{ b }{ x }  }{ \log _{ b }{ a }  }

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PROOF:

RHS\\ \\ =\frac { \log _{ b }{ x }  }{ \log _{ b }{ a }  } \\ \\ =\log _{ b }{ x } \cdot \log _{ a }{ b } \\ \\ =\log _{ a }{ \left( { b }^{ \log _{ b }{ x }  } \right)  } \\ \\ =\log _{ a }{ x } \\ \\ =LHS

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