Prove equality a^(log_b c)=c^(log_b a)

orkesruim40

orkesruim40

Open question

2022-08-19

Prove equality a log b c = c log b a
I'm try to prove the equality:
a log b c = c log b a
I'm having trouble finding information regarding this, also I need to figure out why n log 2 3 is better than 3 log 2 n as a closed form formula for M ( n )? Thank you!

Answer & Explanation

yassou1v

yassou1v

Beginner2022-08-20Added 14 answers

Taking the logarithm to base b of both sides, the equation is equivalent to
( log b c ) ( log b a ) = ( log b a ) ( log b c )   .
I would say n log 2 3 is better because it makes it clear that it is n to a constant power. This is of course also true for 3 log 2 n , but it's not so obvious.
sittesf

sittesf

Beginner2022-08-21Added 1 answers

For the first part, you could use the fact that
log b c = log a c log a b = log a c log b b log b a = ( log a c ) ( log b a )
And then apply it to LHS:
a log b c = a ( log a c ) ( log b a ) = ( a log a c ) log b a = c log b a

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