Question

Number of solution(s) of the equation $\log_2(x^2+3)=\frac{1}{2} \log_{1/3} (\frac{1}{x}+x), x>0$ is -

• A. 0
• B. 1
• C. 2
• D. infinite

Answer 1

Answer: (A)

0

Answer 2

The LHS, $\log_2(x^2+3)$, for $x>0$, has a range of $(\log_2 3, \infty)$, meaning all values are strictly greater than 1. The RHS, $\frac{1}{2} \log_{1/3}(\frac{1}{x}+x)$, simplifies to $\log_3(\frac{1}{\sqrt{x+1/x}})$. For $x>0$, by AM-GM, $x+\frac{1}{x} \ge 2$. Thus, $\frac{1}{\sqrt{x+1/x}} \in (0, \frac{1}{\sqrt{2}}]$. The range of the RHS is $(-\infty, \log_3(\frac{1}{\sqrt{2}})] = (-\infty, -\frac{1}{2}\log_3 2]$, meaning all values are strictly less than 0. Since the ranges do not overlap, there are no solutions.