Prove. I understand that if I flip the base to I'll get . But how do I prove it?

Hello!

We want to prove the identity    (for all and ).

To do this, let's recall what logarithms are. By the definition,   is such a number that    It is known that such a number always exists and is unique (of course for and ).

Therefore to verify this identity we raise to the power of each side:

...

Hello!

We want to prove the identity    (for all and ).

To do this, let's recall what logarithms are. By the definition,   is such a number that    It is known that such a number always exists and is unique (of course for and ).

Therefore to verify this identity we raise to the power of each side:

(the function is one-to one on its domain, therefore this operation gives an equivalent equality).

The left part is equal to by the definition of logarithm, what about the right part? We'll use some properties of powers, and

And this is equal to too, by the definition of logarithm. This way we proved the desired identity.