Question

plz ans this question​

Answer 1

$Given \: \: Question \: \: Is \: \\ \\ log_{2}(x) \times log_{2}( \frac{x}{16} ) = log_{2}( \frac{x}{64} ) \\ find \: \: x \\ \\ Answer \: \\ \\ log_{2}(x )( log_{2}(x) - log_{2}(2 {}^{4} ) ) = log_{2}(x) - log_{2}(2 {}^{6} ) \\ becoz \: \: log( \frac{m}{n} ) = log(m) - log(n) \\ \\ log_{2}(x) ( log_{2}(x) - 4 log_{2}(2) ) = log_{2}(x) - 6 log_{2}(2) \\ becoz \: \: log(x {}^{n} ) = n log(x) \\ \\ log_{2}(x) ( log_{2}(x) - 4) = log_{2}(x) - 6\\ becoz \: \: log_{x}(x) = 1 \\ \\ (log_{2}(x) ) {}^{2} - 4 log_{2}(x) - log_{2}(x) + 6= 0 \\ \\ ( log_{2}(x) ) {}^{2} - 5 log_{2}(x) + 6= 0 \\ \\ put \: \: log_{2}(x) = z \\ \\ z {}^{2} - 5z + 6= 0 \\ \\ z {}^{2} - 3z - 2z + 6 = 0 \\ \\ z(z - 3) - 2(z - 3) = 0 \\ \\ (z - 2) = 0 \: \: \: \: or \: \: \: (z - 3) = 0 \\ \\ z = 2 \: \: \: \: or \: \: \: z = 3 \\ \\ log_{2}(x) = 2 \: \: \: or \: \: \: log_{2}(x) = 3 \\ \\ x = 2 {}^{2} \: \: \: or \: \: \: \:x = 2 {}^{3} \\ becoz \: \: \: IF \: log_{x}(y) = a \: \: \: then \: \: \: y = x {}^{a} \\ \\ therefore \: \: \: x = 4 \: \: \: or \: \: \: x = 8$