The problem is based upon logarithm in which we find the value of an unknown quantity in an equation. And understanding whether the root or (value of x) is real or not and if yes, then how many real roots exist.
The equation \(\log_3 x-\log_x 3 =2\) has
(A) no real solution
(B) exactly one real solution
(C) exactly two real solution
(D) infinitely many real solution.
ISI entrance B. Stat. (Hons.) 2003 problem 2
Logarithm
6 out of 10
challenges and thrills of pre college mathematics
we know when logarithm base and value are interchanged then the whole quantity is equal to the reciprocal of the previous logarithm.
i.e. \( \log_3 x\) = \(\frac{1}{log_x 3}\)
Now we can assume the value of \( \log_3 x\) is \(a\) and the equation will reduce to \(a^2 -1/a=2\) , or \(a^2-2a-1=0\).
An now we can apply Sridharacharya's formula to find the valise(s) of x.
After solving we will get two values of a and they are
\(1\pm \sqrt{2}\)
And now these values will be equal to a or \( \log_3 x\) , And from here we will get two values of x which are real.
So option (C) is the correct option.
I.S.I. & C.M.I. Entrance Program
Taught by students and alumni of I.S.I. & C.M.I.
Publishing original scientific research in a peer-reviewed journal is an achievement many scholars work toward for years. Achieving this feat early in one's academic journey requires exceptional intellectual rigor, creativity, and persistence. Aaditya Punatar and Muskan Sadana have reached this remarkable milestone, successfully publishing their collaborative research paper in the prestigious Journal of Emerging Investigators […]
Her project on quantum-accelerated Steiner trees for 15-minute cities earned a place among the finalists at the world's largest pre-college STEM competition. Prishaa Shrimali of Marymount School of New York, was selected as a Grand Finalist at the 2026 Regeneron International Science and Engineering Fair (ISEF) in Phoenix, Arizona. Her project was Optimizing Urban Accessibility: […]
In 2026, the following Cheenta students have been successful for Indian Statistical Institute's M.Stat Entrance. They ranked within the first 50 in the entire country in these entrances. I.S.I. M.Stat Entrance
In 2026, the following Cheenta students have been successful for Indian Statistical Institute's B.Stat Entrance and Chennai Mathematical Institute's B.Sc. Math Entrance. They ranked within the first 200 in the entire country in these entrances. Most of these students attended the problem solving workshops regularly, which happen 5 days every week. CMI B.Sc. Math Entrance […]