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?

1. no values of m and n can satisfy the given equation
2. m=n
3. 1/m + 1/n = 2
4. 1/m + 1/n = 1
5. 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

---

---

---