Question

plz help me to slove this problem​

Answer 1

$\LARGE\mathfrak{\green{Rules\: Of \: Exponents:-}}$

$\begin{gathered}\boxed{\begin{minipage}{5 cm}\bf{\dag}\:\:\underline{\text{Law of Exponents :}}\\\\\bigstar\:\:\sf\dfrac{a^m}{a^n} = a^{m - n}\\\\\bigstar\:\:\sf{(a^m)^n = a^{mn}}\\\\\bigstar\:\:\sf(a^m)(a^n) = a^{m + n}\\\\\bigstar\:\:\sf\dfrac{1}{a^n} = a^{-n}\\\\\bigstar\:\:\sf\sqrt[\sf n]{\sf a} = (a)^{\dfrac{1}{n}}\end{minipage}}\end{gathered}$

$\LARGE\mathfrak{\pink{Solution:-}}$

$\large\mathbb{\orange{GIVEN:-}}$

${25}^{x - 1} = {5}^{2x - 1} - 100$

$\large\mathbb{\purple{TO \:FIND:-}}$

Value of x .

$\large\mathbb{\red{ACCORDING \:TO\: THE\: QUESTION:-}}$

${25}^{x - 1} = {5}^{2x - 1} - 100 \\ \\ \implies \: {( {5}^{2} )}^{x - 1} = {5}^{2x - 1} - 100 \\ \\ \implies \: {5}^{2x - 2} = {5}^{2x - 1} - 100 \\ \\ \implies \:{5}^{2x - 2} - {5}^{2x - 1} = - 100\\ \\ \implies {5}^{2x - 2} - {5}^{2x - 2} \times {5}^{1} = - 100\\ \\ \implies {5}^{2x - 2}(1 - 5) = - 100\\ \\ \implies {5}^{2x - 2} \times - 4 = - 100\\ \\ \implies {5}^{2x - 2} = \cancel\frac{ - 100}{ - 4} \\ \\ \implies {5}^{2x - 2} = 25\\ \\ \implies {5}^{2x - 2} = {5}^{2} \\ \\ \implies {2x - 2} = 2\\ \\ \implies \: 2x = 2 + 2\\ \\ \implies \: 2x = 4\\ \\ \implies \: x = \cancel\frac{4}{2} \\ \\ \implies \: x = 2$

$\large\mathbb{\purple{ANSWER:-}}$

$\implies \boxed { \pink{x = 2}}$

$\large\mathbb{\blue{VERIFICATION:-}}$

${25}^{x - 1} = {5}^{2x - 1} - 100 \\ \\ \implies \: {25}^{2 - 1} = {5}^{2 \times 2 - 1} - 100\\ \\ \implies \: {25}^{1} = {5}^{3} - 100\\ \\ \implies \: 25 = 125 - 100\\ \\ \implies \:25 = 25(proved)$