If n = 1983 ! , then the value of expression 1 log 2 n + 1 log 3 n + 1 log 4 n + ? + 1 log 1983 n is

Question

If $n = 1983!$ , then the value of expression $\frac{1}{\log_{2} n} + \frac{1}{\log_{3} n} + \frac{1}{\log_{4} n} + \dots + \frac{1}{\log_{1983} n}$ is _________.

Smart Achievers · Accepted Answer

(1)

$\frac{1}{\log_{2} n} + \frac{1}{\log_{3} n} + \frac{1}{\log_{4} n} + \dots + \frac{1}{\log_{1983} n}$

$= \log_{n} 2 + \log_{n} 3 + \dots + \log_{n} 1983$

$= \log_{n} (2 \cdot 3 \cdot 4 \dots 1983) = \log_{n} 1983! = \log_{n} n = 1$

Solution

Q.
If $n = 1983!$ , then the value of expression $\frac{1}{\log_{2} n} + \frac{1}{\log_{3} n} + \frac{1}{\log_{4} n} + \dots + \frac{1}{\log_{1983} n}$ is _________.

Ans.
(1)

$\frac{1}{\log_{2} n} + \frac{1}{\log_{3} n} + \frac{1}{\log_{4} n} + \dots + \frac{1}{\log_{1983} n}$

$= \log_{n} 2 + \log_{n} 3 + \dots + \log_{n} 1983$

$= \log_{n} (2 \cdot 3 \cdot 4 \dots 1983) = \log_{n} 1983! = \log_{n} n = 1$

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Solution

Q. If n=1983!, then the value of expression 1log2n+1log3n+1log4n+⋯+1log1983n is _________.

Ans. (1) 1log2n+1log3n+1log4n+⋯+1log1983n =logn2+logn3+⋯+logn1983 =logn(2·3·4⋯1983)=logn1983!=lognn=1

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Q.
If $n = 1983!$ , then the value of expression $\frac{1}{\log_{2} n} + \frac{1}{\log_{3} n} + \frac{1}{\log_{4} n} + \dots + \frac{1}{\log_{1983} n}$ is _________.

Ans.
(1)

$\frac{1}{\log_{2} n} + \frac{1}{\log_{3} n} + \frac{1}{\log_{4} n} + \dots + \frac{1}{\log_{1983} n}$

$= \log_{n} 2 + \log_{n} 3 + \dots + \log_{n} 1983$

$= \log_{n} (2 \cdot 3 \cdot 4 \dots 1983) = \log_{n} 1983! = \log_{n} n = 1$