Question

if 2log (x+y)=3log3+logx+logy,then show that x/y+y/x=25​

Answer 1

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Answer 2

Solution :-

$\sf 2log(x + y) = 3 log3 + logx + logy \\ \\ \\ \sf \longrightarrow log(x + y)^2 = log3^3 + logx + logy \\ \\ \\ \boxed{ \bf \because m.loga = log a^m } \\ \\ \\ \sf \longrightarrow log(x + y)^2 = log3^3xy \\ \\ \\ \sf \longrightarrow log(x + y)^2 = log27xy \\ \\ \\ \bf \underline{ Cancelling \: log \: on \: both \: sides} \\ \\ \\ \sf \longrightarrow (x + y)^2 = 27xy \\ \\ \\ \sf \longrightarrow x^2 + y^2 + 2xy = 27xy \\ \\ \\ \bf \because (x + y)^2 = {x}^{2} + {y}^{2} + 2xy \\ \\ \bf \underline{Transpose \: 2xy \: to \: RHS} \\ \\ \\ \sf \longrightarrow x^2 + y^2 = 27xy - 2xy \\ \\ \\ \sf \longrightarrow x^2 + y^2 = 25xy$

Divide every term of the equation by xy

$\sf \longrightarrow \dfrac{x^2}{xy} + \dfrac{y^2 }{xy} = \dfrac{25xy}{xy} \\ \\ \\ \sf \longrightarrow \dfrac{x}{y} + \dfrac{y}{x} = 25$

Hence shown.