What is the number of real roots of ( log &#x2061;<!-- ⁡ --> x ) 2 </m

Janet Forbes 2022-07-08 Answered
What is the number of real roots of ( log x ) 2 log x 2 = 0 represents the greatest integer function less than or equal to x.
I know how to solve logarithm equation but due to greatest integer function I am unable to proceed further please help thanks.
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Answers (1)

Wade Atkinson
Answered 2022-07-09 Author has 12 answers
Since [ log x ] log x
we have ( log x ) 2 log x 2 0
This is equivalent to 1 log x 2
When 1 log x 0 , [ log x ] = 1 so that log x = ± 1 If we see that log x = 1 is not in the specified range. Hence log x = 1 and x = 1 10
When 0 log x < 1, [ log x ] = 0 so that log x = ± 2 None of these values in the range.
Similarly we can use 1 log x < 2 this will give us x = 10 3
When log x = 2, [ log x ] = 2 and equation is satisfied. Thus x = 100 is third real root.
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