Prove that:

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

Say that:

\log _{ b }{ x } =p\\ \\ \therefore \quad { b }^{ p }=x

And that:

\log _{ b }{ a } =q\\ \\ \therefore \quad { b }^{ q }=a

Therefore:

\log _{ a }{ \left( x \right) } \\ \\ =\log _{ a }{ \left( { b }^{ p } \right) } \\ \\ =p\log _{ a }{ \left( b \right) } \\ \\ =p\cdot \frac { 1 }{ \log _{ b }{ a } } \\ \\ =\frac { \log _{ b }{ x } }{ \log _{ b }{ a } }

Other posts you may be interested in...