What is the value of the expression &1log_2 100!+1log_3 100! & +1log_4 100!+… & +1log_100100!?

1998 AMC 12 Problem 22

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22.

What is the value of the expression

\begin{aligned} &\frac1{\log_2 100!}+\frac1{\log_3 100!}\\ &\quad+\frac1{\log_4 100!}+\cdots\\ &\qquad+\frac1{\log_{100}100!}? \end{aligned}

0.01

0.1

1

2

10

Use the reciprocal identity

\frac{1}{\log_bx}=\log_xb

Combine the resulting logarithms of

2,3,\ldots,100

Each term equals

\log_{100!}b.

Therefore the sum is

\begin{aligned} &\log_{100!}(2\cdot3\cdots100)\\ &\qquad=\log_{100!}(100!)=1. \end{aligned}

Thus C is correct.