Quantitative Aptitude – Logarithm – If log4m + log4n
XAT 2021 Exam Paper – Quantitative Aptitude – Logarithm – If log4m + log4n
Q. If log(base 4)m + log(base 4)n = log(base 2)(m+n), where m and n are positive real numbers, then which of the following must be true?
no values of m and n can satisfy the given equation
m=n
1/m + 1/n = 2
1/m + 1/n = 1
m^2 + n^2 = 1
Answer: no values of m and n can satisfy the given equation
Solutions:
Given,
log(base 4)m+ log(base 4)n=log(base 2)(m+n)
Or log(base 4)(mn)=log(base 2)(m+n)
or log(base (2^2 ))(mn)=log(base 2)(m+n)
1/2 log(base 2)(mn)=log(base 2)(m+n)
log(base 2)(mn)^(1/2) =log(base 2)(m+n)
mn^(1/2)=m+n ———1)
As we know AM>= GM
So (m+n)/2 >= mn^(1/2)
Thus there is no value that can satisfy the above equation