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\} $