Question

The number of solutions of the equation
$5^x+5^{-x}={\log }_{10}25,x\in R$ is

• A. 0
• B. 1
• C. 2
• D. infinitely many

Answer 1

Answer: (A)

0

$5^x+5^{-x}=lo{g}_{10}25$
Let $5^x=t$
$t+\frac{1}{t}=lo{g}_{10}25$

$t+\frac{1}{t}\ge 2$ for all $t$ as A.M $\ge$G.M

but $lo{g}_{10}25<2$
$\therefore$ L.H.S. $\ne$ R.H.S
$\Rightarrow$ No roots