Question

How does adding the log together automatically mean that it is a factorial?

Answer 1

Answer: The answer is given below.

Step-by-step explanation: We are given an equality involving logarithm and we are to show the implication of L.H.S. to R.H.S.

We will be using the following two properties of logarithm:

`(i)~\log_ba=(1)/(\log_ab),\\\\\\(ii)~log_ab+\log_ac=\log_a(bc).`

The proof is as follows:

`L.H.S.\\\\\\=(1)/(\log_2N)+(1)/(\log_3N)+(1)/(\log_4N)+\cdots+(1)/(\log_(100)N)\\\\\\=\log_N2+\logN3+\log_N4+\cdots+\log_N100\\\\=\log_N\{2.3.4...100\}\\\\=\log_N\{1.2.3.4...100\}\\\\=\log_N{100!}\\\\=(1)/(\log_(100!)N)\\\\=R.H.S.`

Hence proved.