213k views
1 vote
How does adding the log together automatically mean that it is a factorial?

How does adding the log together automatically mean that it is a factorial?-example-1
User Dasup
by
8.7k points

1 Answer

4 votes

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.

User Jco
by
7.8k points

No related questions found

Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories