Question

(3/6)^6 x (16/9)^5=(4/3)^x-3

Answer 1

GIVEN :–

$\\ \implies{ \bold{ { \left( \frac{3}{6} \right) }^{6} \times { \left( \frac{16}{9} \right) }^{5} = \left( \dfrac{4}{3} \right)^{x - 3} }}$

TO FIND :–

• Value of 'x' = ?

SOLUTION :–

$\\ \implies{ \bold{ { \left( \dfrac{3}{6} \right) }^{6} \times { \left( \dfrac{16}{9} \right) }^{5} = \left( \dfrac{4}{3} \right)^{x - 3} }}$

$\\ \implies{ \bold{ { \left( \dfrac{1}{2} \right) }^{6} \times { \left( \dfrac{16}{9} \right) }^{5} = \left( \dfrac{4}{3} \right)^{x - 3} }}$

• We should write this as –

$\\ \implies{ \bold{ { \left( \dfrac{1}{2} \right) }^{6} \times { \left( \dfrac{ {2}^{4} }{ {3}^{2} } \right) }^{5} = \left( \dfrac{4}{3} \right)^{x - 3} }}$

$\\ \implies{ \bold{ { \left( \dfrac{1}{2} \right) }^{6} \times { \left( \dfrac{ {2}^{20} }{ {3}^{10} } \right) } = \left( \dfrac{4}{3} \right)^{x - 3} }}$

$\\ \implies{ \bold{ { \left( \dfrac{ {2}^{20 - 6} }{ {3}^{10} } \right) } = \left( \dfrac{4}{3} \right)^{x - 3} }}$

$\\ \implies{ \bold{ { \left( \dfrac{ {2}^{14} }{ {3}^{10} } \right) } = \left( \dfrac{4}{3} \right)^{x - 3} }}$

$\\ \implies{ \bold{ { \left( \dfrac{ {4}^{7} }{ {3}^{10} } \right) } = \left( \dfrac{4}{3} \right)^{x - 3} }}$

• Let's take log on both sides –

$\\ \implies{ \bold{ {log{ \left( \dfrac{ {4}^{7} }{ {3}^{10} } \right) }} =(x - 3) log{ \left( \dfrac{4}{3} \right) }}}$

$\\ \implies{ \bold{ \dfrac{ {log{ \left( \dfrac{ {4}^{7} }{ {3}^{10} } \right) }}}{log{ \left( \dfrac{4}{3} \right) }} =(x - 3) }}$

$\\ \implies{ \bold{log{ \left( \dfrac{ {4}^{7} }{ {3}^{10} } - \dfrac{4}{3} \right)} =(x - 3) }}$

$\\ \implies \large{ \boxed{ \bold{x = 3 + log{ \left( \dfrac{ {4}^{7} }{ {3}^{10} } - \dfrac{4}{3} \right)}}}}$

Answer 2

Step-by-step explanation:

${\red{\underline{\underline{\bold{Given:-}}}}}$

• $( { \frac{3}{6}) }^{6} \times { (\frac{16}{9}) }^{5} = { (\frac{4}{3}) }^{x - 3}$

${\blue{\underline{\underline{\bold{To\:Find:-}}}}}$

• The value of x

${\green{\underline{\underline{\bold{Solution:-}}}}}$

${ (\frac{3}{6} )}^{6} \times {(\frac{16}{9} )}^{5} = { (\frac{4}{3}) }^{x - 3} \\ \\ \implies { (\frac{1}{2}) }^{6} \times { (\frac{ {2}^{4} }{ {3}^{2} }) }^{5} = {(\frac{4}{3} )}^{x - 3} \\ \\ \implies { (\frac{1}{2} )}^{6} \times \frac{ {2}^{20} }{ {3}^{10} } = { (\frac{4}{3} )}^{ x - 3} \\ \\ \implies \frac{ {2}^{20 - 6} }{ {3}^{10} } = { (\frac{4}{3})}^{x - 3} \\ \\ \implies \frac{ {2}^{14} }{ {3}^{10} } = {( \frac{4}{3} )}^{x - 3} \\ \\ \implies \frac{ {4}^{7} }{ {3}^{10} } = { (\frac{4}{3}) }^{x - 3}$

Applying logarithm on both sides:-

$log( \frac{ {4}^{7} }{ {3}^{10} } ) = log( { (\frac{4}{3} )}^{x - 3} ) \\ \\ \implies log( \frac{ {4}^{7} }{ {3}^{10} } ) = (x - 3) log( \frac{4}{3} ) \\ \\ \implies log( \frac{ {4}^{7} }{ {3}^{10} } - \frac{4}{3} ) = ( x - 3) \\ \\ \implies3 + log( \frac{ {4}^{7} }{ {3}^{10} } - \frac{4}{3} ) = x$

Therefore:-

$\boxed{x =3 + log( \frac{ {4}^{7} }{ {3}^{10} } - \frac{4}{3} ) }$