We have, 52x−24.5x−25=0⇒5x2−245x−25=0⇒a2−24a−25=0,a=5x⇒(a−25)(a+1)=0⇒a=25,−1When a=25⇒5x=52⇒x=2When a=−1⇒5x=−1⇒x=ϕHence the solution set is {2}.

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$5^{2 x} - {24.5}^{x} - 25 = 0$

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Detailed Solution

We have,

$\begin{array}{l} 5^{2 x} - {24.5}^{x} - 25 = 0 \\ \Rightarrow {(5^{x})}^{2} - 24 (5^{x}) - 25 = 0 \\ \Rightarrow a^{2} - 24 a - 25 = 0, a = 5^{x} \\ \Rightarrow (a - 25) (a + 1) = 0 \\ \Rightarrow a = 25, - 1 \end{array}$

When $a = 25 \Rightarrow 5 x = 5^{2} \Rightarrow x = 2$

When $a = - 1 \Rightarrow 5 x = - 1 \Rightarrow x = ϕ$

Hence the solution set is {2}.

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