Logarithmic Proof (4)

Prove that:

\log _{ a }{ \left( \frac { x }{ p }  \right)  } =\log _{ a }{ \left( x \right)  } -\log _{ a }{ \left( p \right)  }

Say that:

\log _{ a }{ \left( x \right)  } =m\\ \\ \therefore \quad { a }^{ m }=x

And that:

\log _{ a }{ \left( p \right)  } =n\\ \\ \therefore \quad { a }^{ n }=p

Therefore:

\log _{ a }{ \left( \frac { x }{ p }  \right)  } \\ \\ =\log _{ a }{ \left( \frac { { a }^{ m } }{ { a }^{ n } }  \right)  } \\ \\ =\log _{ a }{ \left( { a }^{ \left( m-n \right)  } \right)  } \\ \\ =\left( m-n \right) \log _{ a }{ \left( a \right)  } \\ \\ =m\log _{ a }{ \left( a \right)  } -n\log _{ a }{ \left( a \right)  } \\ \\ =\log _{ a }{ \left( { a }^{ m } \right) -\log _{ a }{ \left( { a }^{ n } \right)  }  } \\ \\ =\log _{ a }{ \left( x \right)  } -\log _{ a }{ \left( p \right)  }

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