Log of many Logs How can I compute the values of n for which the following expression exists? log_e( log_e( log_e( log_e(... log_e(n)))) It is for instance apparent that when n=e, the second application of log_e is undefined.

solvarmedw

solvarmedw

Answered question

2022-09-01

Log of many Logs
How can I compute the values of n for which the following expression exists?
log e ( log e ( log e ( log e ( log e ( n ) ) ) )
It is for instance apparent that when n = e, the second application of log e is undefined.

Answer & Explanation

Kaleb Harrell

Kaleb Harrell

Beginner2022-09-02Added 14 answers

As ln x is apriori defined only for x > 0, you need
n > e e e e e 0
where the tower has as many es as your expression has log e s
Gunsaz

Gunsaz

Beginner2022-09-03Added 3 answers

You have the sequence
0 , 1 , e , e e , e e e , e e e e , ,
for which for the nth member (starting with 0), you can apply log only n times before you end up with log 0 . The sequence diverges, so there is no point at which you can take as many logs as you like.

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