Question

The number of solution of the equation $2-x+3{\log }_{5}2={\log }_5\left(3^x-5^{2-x}\right)$ is

• A. 1
• B. 2
• C. 3
• D. 5

Answer 1

Answer: (A)

$2-x+3{\log }_{5}2={\log }_5\left(3^x-5^{2-x}\right)$
${\log }_5{5}^{2-x}+{\log }_5{2}^3={\log }_5\left(3^x-5^{2-x}\right)$
$5^{2-x}\cdot 8=3^x-5^{2-x}$
$5^{2-x}(8+1)=3^x$
$\frac{5^2}{5^x}\cdot 9=3^x$
$15^2=15^x\Rightarrow x=2$