Question

Find x in log2 256 x²= 1+2log2(1/2x⁴​)

Answer 1

$\large\underline{\sf{Given- }}$

$\rm :\longmapsto\: log_{2}(256 {x}^{2} ) = 1 + 2 log_{2} \bigg( \dfrac{1}{2 {x}^{4} } \bigg)$

$\large\underline{\sf{To\:Find - }}$

$\rm :\longmapsto\:value \: of \: x$

$\large\underline{\sf{Solution-}}$

Given that

$\rm :\longmapsto\: log_{2}(256 {x}^{2} ) = 1 + 2 log_{2} \bigg( \dfrac{1}{2 {x}^{4} } \bigg)$

$\rm :\longmapsto\: log_{2}(256 {x}^{2} ) = log_{2}(2) + 2 log_{2} \bigg( \dfrac{1}{2 {x}^{4} } \bigg)$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \green{\boxed{ \bf \because \: log_{a}(a) = 1 }}$

$\rm :\longmapsto\: log_{2}(256 {x}^{2} ) = log_{2}(2) + log_{2} \bigg( \dfrac{1}{2 {x}^{4} } \bigg)^{2}$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \green{\boxed{ \bf \because \: log( {x}^{y} ) = y log(x) }}$

$\rm :\longmapsto\: log_{2}(256 {x}^{2} ) = log_{2}(2) + log_{2} \bigg( \dfrac{1}{4 {x}^{8} } \bigg)$

$\rm :\longmapsto\: log_{2}(256 {x}^{2} ) = log_{2} \bigg(2 \times \dfrac{1}{4 {x}^{8} } \bigg)$

$\: \: \: \: \: \: \: \green{\boxed{ \bf \because \: log_{a}(x) + log_{a}(y) = log_{a}(xy)}}$

$\rm :\longmapsto\: log_{2}(256 {x}^{2} ) = log_{2} \bigg(\dfrac{1}{2{x}^{8} } \bigg)$

$\rm :\longmapsto\: 256 {x}^{2} = \dfrac{1}{2{x}^{8} }$

$\rm :\longmapsto\: {x}^{10} = \dfrac{1}{512}$

$\rm :\longmapsto\: {x}^{10} = \dfrac{1}{ {2}^{9} }$

$\bf\implies \:x \: = \: \pm \: \bigg(\dfrac{1}{2} \bigg)^{ \dfrac{9}{10} }$

Additional information :-

$\rm :\longmapsto\: log(xy) = logx \: + \: logy$

$\rm :\longmapsto\: log_{x}(y) = \dfrac{logy}{logx}$

$\rm :\longmapsto\: log_{x}(y) = \dfrac{1}{ log_{y}(x) }$

$\rm :\longmapsto\: log_{ {x}^{p} } {x}^{q} = \dfrac{q}{p}$