Question

Find the roots of the equation in ratio form?? (jee Mains) plz answer this fast urgent whoever gives the best answer I will him/her brainliest

Answer 1

Given,

$x = \pi ^{4\log_x \pi} \\ \\ \text{We know that,} \ {\log_a b} = \frac{\log b}{\log a} \\ \\ Similarly ,log_x \pi = \frac{\log \pi}{\log x} \\ \\ Threfore, \\ \\ x = \pi^{4\frac{\log \pi}{\log x}} \\ \\ \text{Taking log of both sides} \\ \\ \log x = 4 \frac{\log \pi}{\log x} \ \log \pi \\ \\ (\log x)^{2} = 2* \log \pi * 2 \log \pi \\ \\ (\log x)^{2} = \log \pi^{2} \log \pi^{2} \\ \\ (\log x)^{2} = (\log \pi^{2})^{2} \\ \log x = \frac{+}{} \log \pi^{2} \\ \\ \text{We will observe two values of x, Let first be} \ x_1 \text{and second be} \ x_2 \\ \\ \textbf{Taking +ve} \\ \\ \log x_1 = \log \pi^{2} \\ \\ x_1 = \pi^{2} \\ \\ \text{Taking -ve} \\ \\ \log x_2 = - \log \pi^{2} \\ \\ \log x_2 = \log \frac{1}{\pi^{2}} \\ \\ x_2 = \frac{1}{\pi^{2}} \\ \\ Hence, \\ \\ \frac{x_1}{x_2} = \frac{\pi^{4}}{1} \\ \\ \textbf{Therefore, Ratio of roots is} \ 1: \pi^{4}$