Question

${25}^{x - 1} + 100 = {5}^{2x - 1}$
Find the value of x

Answer 1

Answer :

$given \: equation \: = {25}^{x - 1} + 100 = {5}^{2x - 1} \\ \\ = > \: firstly \: move \: the \: constant \: to \: ther \: right \: side \: \\ \\ = > \: {25}^{x - 1} = {5}^{2x - 1} - 100 \\ \\ = > \: move \: the \: expression \: {5}^{2x - 1} \: to \: left \: side \: \\ \\ = > \: {25}^{x - 1} - {5}^{2 x- 1} = - 100 \\ \\ = > \: write \: the \: expression \: in \: the \: exponental \: form \: with \: base \: 5 \\ \\ = > \: {5}^{2 x- 2} - {5}^{2x - 1} = - 100 \\ \\ = > \: factor \: the \: expression \\ \\ = > \: (1 - 5)5 {}^{2x - 2} = - 100 \\ \\ = > \: calculate \: the \: difference \\ \\ = > \: - 4 \times {5}^{2x - 2} = - 100 \\ \\ = > \: divide \: both \: sides \: by \: 4 \\ \\ = > \: {5}^{2x - 2} = 25 \\ \\ = > \: write \: the \: number \: in \: the \: exponental \: form \: with \: base \: 5 \\ \\ = > \: {5}^{2x - 2} = {5}^{2} \\ \\ = > \: bases \: are \: same \: so \: set \: the \: exponents \: equal \\ \\ = > \: 2x - 2 = 2 \\ \\ = > \: 2x = 2 + 2 \\ \\ = > \: 2x = 4 \\ \\ = > \: x = \frac{4}{2} \\ \\ = > \: x = 2$



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