Question

if log x to the base 3 is equal to a and log x to the base 7 is equal to b then find log x to the base 21​

Answer 1

$HEYA \: \\ \\ \\ log_{3}(x) = a \: \: \: \: \: \: \: \: \: \: \: \: log_{7}(x) = b \\ \\ 1 \div ( log_{x}(3) ) = a \\and \\ 1 \div (log_{x}(7) ) = b \: \: \: \: \: by \: using \: base \: change \: formula \: \\ \\ log_{x}(3) = 1 \div a \: \: \: \: .....(i) \\ \\ log_{x}(7) = 1 \div b \: \: \: \: .....(ii) \\ \\ add \: both \: the \: equations \: we \: have \\ \\ log_{x}(3) + log_{x}(7) = (a + b) \div ab \\ \\ log_{x}(7 \times 3) = (a + b) \div ab \\ becoz \: \: log(m) + log(n) = log(m \times n) \\ \\ log_{x}(21) = (a + b) \div ab \\ \\ log_{21}(x) = (ab) \div (a + b)$