No Calculus!

Algebra Level 4

A small little kid was giving this question, without the use of calculus, can you help her solve the problem below?

For x > 1 x>1 , evaluate the minimum value of

1 ln ⁡ x + 3 + ln ⁡ x \LARGE \frac {1}{\ln x} + 3 + {\ln x}


The answer is 5.

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Narayan Prasad by Narayan Prasad , 29, India

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2 solutions

Akshaj Garg
Aug 25, 2019

Applying AM-GM inequality to ln ⁡ x + 1 ln ⁡ x \ln x+\frac{1}{\ln x} we get minimum value as 2. Adding it to 3 gives the answer 5 \boxed{5}

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Prakhar Bindal
Mar 7, 2015

Simply As x>1 so lnx would be positive as the number and base of logarithm are on the same side of unity.and sum of two positive reciprocal quantities is always greater than equal to 2 . so minimum value 2+3 = 5 when x = e(euler's number or constant)

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