Simplifying this logarithm series ∑ i = 2 99 1 log i ⁡ ( 99...

Savanah Boone

Savanah Boone

Answered

2022-07-01

Simplifying this logarithm series
i = 2 99 1 log i ( 99 ! )
How would you evaluate (or at least simplify) this logarithm series?

Answer & Explanation

isscacabby17

isscacabby17

Expert

2022-07-02Added 13 answers

Note that log a b = log b log a , for any choice of base of the logarithm. (Mathematicians tend to mean the natural logarithm when they write log.)
In particular, 1 log a b = log a log b = log b a.
Therefore your sum is:
i = 2 99 log 99 ! i = log 99 ! ( i = 2 99 i ) = log 99 ! 99 ! = 1
where in first equality I used the well-known fact that log = log .

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