Logarithmic Proof (2)

Prove that:

\log _{ a }{ (xp)=\log _{ a }{ (x)+\log _{ a }{ (p) }  }  }

Say that:

\log _{ a }{ x } =m\\ \\ \therefore \quad { a }^{ m }=x

And say that:

\log _{ a }{ p } =n\\ \\ \therefore \quad { a }^{ n }=p

Therefore:

\log _{ a }{ (xp) } \\ \\ =\log _{ a }{ ({ a }^{ m }\cdot { a }^{ n }) } \\ \\ =\log _{ a }{ ({ a }^{ (m+n) }) } \\ \\ =(m+n)\log _{ a }{ (a) } \\ \\ =m\log _{ a }{ (a)+n\log _{ a }{ (a) }  } \\ \\ =\log _{ a }{ ({ a }^{ m })+\log _{ a }{ ({ a }^{ n }) }  } \\ \\ =\log _{ a }{ (x)+\log _{ a }{ (p) }  }

Other posts you may be interested in...