Question

Sum of all values of $x$ lying between $0\&2\pi$ and satisfying the equation $2{\log }_{25}\left(5^{2\cos x}-4\right)=2\cos x+\cos \pi$ is

• A. $\frac{\pi }{2}$
• B. $2\pi$
• C. $\frac{\pi }{3}$
• D. $\frac{5\pi }{3}$

Answer 1

Answer: (B)

${\log }_5\left(5^{2\cos x}-4\right)=2\cos x-1$
$5^{2\cos x}-4=\frac{5^{2\cos x}}{5}$
let $5^{2\cos x}=y$ we get
$5(y-4)=y\Rightarrow y=5=5^{2\cos x}$
$\Rightarrow \cos x=\frac{1}{2}\Rightarrow x\Rightarrow \left\{\frac{\pi }{3},\frac{5\pi }{3}\right\}$