How to show that n^(ln(ln(n))) = ln(n)^(ln(n))

Barrett Osborn

Barrett Osborn

Answered question

2022-11-03

How to show that n ln ( ln ( n ) ) = ln ( n ) ln ( n )
I have verified that n ln ( ln ( n ) ) = ln ( n ) ln ( n ) by plugging in values for n, but do not understand why it is true. I am not aware of any log rules that can be used to simplify n ln ( ln ( n ) )

Answer & Explanation

Kayleigh Cross

Kayleigh Cross

Beginner2022-11-04Added 19 answers

Note that n = e ln n so
n ln ( ln n ) = e ln n ln ( ln n ) = ( ln n ) ln n
Clara Dennis

Clara Dennis

Beginner2022-11-05Added 5 answers

Notice that if we can prove
a log b c = c log b a
then we have proved your problem with a = n, b = e, and c = ln n
We now do just that. Let a , b , c be arbitrary real numbers. Notice that b log b a = a. Also notice that ( x y ) z = x y z for real numbers x , y , z. With these two facts in hand we have that
a log b c = ( b log b a ) log b c = b ( log b a ) ( log b c ) = b ( log b c ) ( log b a ) = ( b log b c ) log b a = c log b a

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