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

Nikolai Decker 2022-10-11 Answered
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?
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Answers (2)

Amadek6
Answered 2022-10-12 Author has 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
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benatudq
Answered 2022-10-13 Author has 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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