Logarithm formula proof Prove: x^(log(y))=y^(log(x))

Nikolai Decker

Nikolai Decker

Answered question

2022-10-11

Logarithm formula proof
Prove:
x log ( y ) = y log ( x )
I have been trying this for the past 1 hour, still cant prove it.
I started with
log b ( y ) = m
log b ( x ) = n
To show:
x m = y n
How do i proceed?

Answer & Explanation

Amadek6

Amadek6

Beginner2022-10-12Added 21 answers

Start with the following equality:
log ( x ) log ( y ) = log ( x ) log ( y ) e log ( x ) log ( y ) = e log ( x ) log ( y ) ( e log x ) log y = ( e log y ) log x x log y = y log x
Identities used:
e log x = x
( a m ) n = ( a n ) m = a m n
benatudq

benatudq

Beginner2022-10-13Added 2 answers

Since we can take logs on both sides because taking the log preserves the equality i.e.
log a = log b a = b
in short, we can still prove the statement if are working the logarithm of the equality.
Anyway,
log ( x log y ) = log ( y ) log ( x )
similarly
log ( y log x ) = log ( x ) log ( y )
here I used log a b = b log a.

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