I Love Logarithms!

Algebra Level 2

If n = 1 × 2 × 3 × . . . × m n=1\times2\times3\times...\times m for m ∈ Z + , m > 2 m\in\mathbb{Z}^{+},~m>2 then find the value of ∑ r = 1 m 1 log ⁡ r n \displaystyle\sum_{r=1}^{m}\frac{1}{\log_{r}{n}}


The answer is 1.

Omkar Kulkarni by Omkar Kulkarni , 22, India

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1 solution

Sarthak Rath
Apr 14, 2015

the expression can be written as : l o g n 1 log_n 1 + l o g n 2 log_n 2 + l o g n 3 log_n 3 ......... + l o g n m log_n m

which is nothing but :

l o g n m ! log_n m!

or

l o g n n log_n n = 1 \boxed{1}

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