Question

if log x^3 y^4= a, log xy=b find log x and log y

Answer 1

$Given \: \\ \\ log_{}(x {}^{3} y {}^{4} ) = a \: \: \: \: and \: \: \: log(xy) = b \: \\ find \: \: \: log(x) \: \: and \: \: \: log(y) \\ \\ Answer \: \\ \\ log(x {}^{3} y {}^{4} ) = a \\ \\ log((xy) {}^{3} y) = a \\ \\ log(xy) {}^{3} + log(y) = a \\ becoz \: \: \: log(m) + log(n) = log(mn) \\ \\ 3 log(xy) + log(y) = a \\ becoz \: \: \: log(x {}^{n} ) = n log(x) \\ \\ 3b + log(y) = a \\ becoz \: \: it \: \: is \: \: given \: \: \: log(xy) = b \\ \\ log(y) = a - 3b \\ \\ log(xy) = b \: \: \: (given) \\ \\ log(x) + log(y) = b \\ \\ log(x) + (a - 3b) = b \\ \\ log(x) =( 4b - a )\\ \\ therefore \: \: \\ log(x) = (4b - a) \: \: \: and \: \: \: log(y) = (a - 3b)$