Question

If 3 + log x base 10 is equal to 2 log y base 10 then find the value of x in terms of y

Answer 1

$Given \: \: Question \: Is \: \\ \\ 3 + log_{10}(x) = 2 log_{10}(y) \\ find \: \: x \: \: in \: terms \: \: of \: \: y \\ \\ Answer \: \\ \\ 3 + log_{10}(x) = 2 log_{10}(y) \\ \\ 3 + log_{10}(x) = log_{10}(y {}^{2} ) \\ becoz \: \: log(n {}^{p} ) = p log(n) \: \\ \\ 3 + log_{10}(x) - log_{10}(y {}^{2} ) = 0 \\ \\ 3 + log_{10}( \frac{x}{y {}^{2} } ) = 0 \\ becoz \: \: log(m) - log(n) = log( \frac{m}{n} ) \\ \\ log_{10}( \frac{x}{y {}^{2} } ) = - 3 \\ \\ \frac{x}{y {}^{2} } = {10}^{ - 3} \\ becoz \: \: if \: \: log_{t}(r) = d \: \: then \: \: r = {t}^{d} \\ \\ x = 10 {}^{ - 3} y {}^{2} \\ \\ x = \frac{y {}^{2} }{10 {}^{3} } \\ \\ 10 {}^{3} x = y {}^{2} \\ \\ y = \sqrt{10 {}^{3} x} \: \: \: \: \: or \: \: \: \: \: y = - \sqrt{10 {}^{3} x} \\ \\ therefore \: the \: required \: relationship \: between \: \\ x \: and \: y \: is \\ \\ y = - \sqrt{10 {}^{3} x} \: \: \: \: \: \: or \: \: \: \: \: y = \sqrt{10 {}^{3}x }$