What is the value of the expression

$\frac{1}{\log_{2}100!}+\frac{1}{\log_{3}100!}+\frac{1}{\log_{4}100!}+\cdots+\frac{1}{\log_{100}100!}$

2
1
.01
10

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1) Any log with base x and number y can be written as [log(y)/log(x)]

2) if you write the numbers in question in the above manner you will see the numerator as [log2 +log3 + log4+....log100

3) denominator will be common log 100!

4) logA + logB + logC +...logZ = log( A

BC....Z)5) therefore numerator = log (2

345.....100) = log (100!)6)therefore its log (100!)/ Log(100!) = 1