Question

solve the below question 2^2x-3=5^x-1​

Answer 1

Solution :-

${2}^{2x - 3} = {5}^{x - 1}$

Taking log on both sides

$\implies log {2}^{2x - 3} = log{5}^{x - 1}$

$\implies (2x - 3)log2= (x -1) log5$

[ Because, by power rule, log a^m = mlog a ]

$\implies 2xlog2 - 3log2= x log5 + log5$

$\implies 2xlog2 - xlog5= log5 + 3log2$

$\implies x(2log2 - log5)= log5 + 3log2$

$\implies x(log2^{2} - log5)= log5 + log2^{3}$

[ Because, by power rule, log a^m = mlog a ]

$\implies x(log4 - log5)= log5 + log8$

$\implies x \bigg(log \dfrac{4}{5} \bigg)= log(5 \times 8)$

[ Because, log a - log b = log a/b , log a + log b = log ab ]

$\implies x \bigg(log \dfrac{4}{5} \bigg)= log40$

$\implies x= \dfrac{log40}{log \dfrac{4}{5} }$

Therefore the value of x is log40/( log 4/5 )