Question

log 3 base x log 3 base(x/81) = log 3 base (x/729)​

Answer 1

$log_{x}(3) . log_{ \frac{x}{81} }(3) = log_{ \frac{x}{729} }(3)$

$\frac{1}{ log_{3}(x) } . \frac{1}{ log_{3}( \frac{x}{81} ) } = \frac{1}{ log_{3}( \frac{x}{729} ) }$

$\frac{1}{ log_{3}(x) } . \frac{1}{ log_{3}(x) - log_{3}(81) } = \frac{1}{ log_{3}(x) - log_{ 3 }(729) }$

$\frac{1}{ log_{3}(x) } . \frac{1}{ log_{3}(x) - 4 } = \frac{1}{ log_{3}(x) - 6}$

$\frac{1}{ log_{3} ^{2} (x) - 4 log_{3}(x) } = \frac{1}{ log_{3}(x) - 6}$

$log_{3} ^{2} (x) - 5 log_{3}(x) + 6 = 0$

$log_{3}(x) = 3 \: \: \: or \: \: \: log_{3}(x) = 2$

$∴ x = 27 \: \: \: or \: \: \: 9$