An algebra problem by Mateus Gomes

Algebra Level 3

$\dfrac{1}{\log_2(2010!)}+\dfrac{1}{\log_3(2010!)}+\cdots+\dfrac{1}{\log_{2009}(2010!)}+\dfrac{1}{\log_{2010}(2010!)} = \, ?$

Notation : $!$ denotes the factorial notation. For example, $8! = 1\times2\times3\times\cdots\times8$ .

1 solution

Zee Ell
Aug 19, 2016

$\text {Due to the change of base theorem: }$

$\frac {1}{log_k(2010!)} = \frac {1}{\frac {ln(2010!)}{ln(k)}} = \frac {ln(k)}{ln(2010!)} = log_{2010!}(k)$

$\text {Therefore, if we apply this for k = 2, 3, ... , 2009, 2010}$

$\text {(and (for the sake of aesthetics) we add } 0 = log_{2010!}(1) \text { ), we get: }$

$S = log_{2010!}(1) + log_{2010!}(2) + ... + log_{2010!}(2009) + log_{2010!}(2010) =$

$= log_{2010!}(1 × 2 × ... × 2009 × 2010) = log_{2010!}(2010!) = \boxed {1}$