# Tag Archives: proofs

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

# Real Mathematical Proofs & Philosophical Videos

Dear users,

I’ve set up a Pinterest account that you can now explore. If you visit this link https://www.pinterest.com/maths_videos/, you will find plenty of maths proofs that I’ve created – including videos related to the simulation hypothesis, information theory, black holes, space-time and topology.

Hope to see you there! 🙂 # 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: $\log _{ a }{ \left( \frac { x }{ p } \right) } \\ \\ =\log _{ a }{ \left( \frac { { a }^{ m } }{ { a }^{ n } } \right) } \\ \\ =\log _{ a }{ \left( { a }^{ \left( m-n \right) } \right) } \\ \\ =\left( m-n \right) \log _{ a }{ \left( a \right) } \\ \\ =m\log _{ a }{ \left( a \right) } -n\log _{ a }{ \left( a \right) } \\ \\ =\log _{ a }{ \left( { a }^{ m } \right) -\log _{ a }{ \left( { a }^{ n } \right) } } \\ \\ =\log _{ a }{ \left( x \right) } -\log _{ a }{ \left( p \right) }$

# Logarithmic Proof (3)

Prove that: $\log _{ a }{ \left( { a }^{ n } \right) =n\log _{ a }{ (a) } }$

Say that: ${ a }^{ n }=p$

Therefore: $\log _{ a }{ p=n }$

So: $\log _{ a }{ \left( { a }^{ n } \right) } \\ \\ =\log _{ a }{ \left( p \right) } \\ \\ =\log _{ a }{ \left( p \right) } \cdot 1\\ \\ =\log _{ a }{ \left( p \right) \cdot \log _{ a }{ \left( a \right) } } \\ \\ =n\cdot \log _{ a }{ \left( a \right) }$

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

# Logarithmic Proof (1)

Prove that: $\log _{ a }{ x=\frac { 1 }{ \log _{ x }{ a } } }$

Proof: $\log _{ a }{ x } =p\\ \\ { a }^{ p }=x\\ \\ { x }^{ \frac { 1 }{ p } }=a\\ \\ \log _{ x }{ a } =\frac { 1 }{ p } \\ \\ p\log _{ x }{ a } =1\\ \\ p=\frac { 1 }{ \log _{ x }{ a } } \\ \\ As,\quad p=p:\\ \\ \log _{ a }{ x } =\frac { 1 }{ \log _{ x }{ a } }$