Question

Solve the following :-

$\sf \dfrac{ {81}^{ \bigg(\dfrac{1}{ log_{5}(9) } \bigg)} + {3}^{ \bigg( \dfrac{3}{ log_{ \sqrt{6} }(3) } \bigg)} }{409} \times \bigg[ {(7)}^{ \bigg(\dfrac{2}{ log_{25}(7) } \bigg) } - {(125)}^{( log_{25}(6) )} \bigg ]$

Topic - Logarithm

Class - 11th

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Answer 1

$\bigstar\large {\pmb{ \sf Question : - }}$

$\sf \dfrac{ {81}^{ \bigg(\dfrac{1}{ log_{5}(9) } \bigg)} + {3}^{ \bigg( \dfrac{3}{ log_{ \sqrt{6} }(3) } \bigg)} }{409} \times \bigg[ {(7)}^{ \bigg(\dfrac{2}{ log_{25}(7) } \bigg) } - {(125)}^{( log_{25}(6) )} \bigg ]$

$\bigstar\large {\pmb{ \sf Solution: - }}$

• There are 4 terms in this expression
• We will be solving all of them individually.

Let's get started :D

$\maltese \: \sf {81}^{ \bigg( \dfrac{1}{ log_{5}(9) } \bigg)}$

we know that :-

$\leadsto \large \underline{ \boxed {\pink{{ \frak{ \dfrac{1}{ log_{b}(a) } = log_{a}(b) }}}}} \bigstar$

$\dag \: \underline \frak{applying \: the \: same \: concept \: for \: the \: equation : - }$

$: \implies \large \: \sf {81}^{ \bigg( \dfrac{1}{ log_{5}(9) } \bigg)} = { {9}^{2} }^{ (log_{9}(5)) }$

$: \implies \large\sf {9}^{ log_{9}( {5}^{2} ) }$

According to the following property:-

$\leadsto \large \underline{ \boxed {\pink{{ \frak{ {a}^{ log_{a}(b) } = b }}}}} \bigstar$

$:\implies \large \sf {9}^{( log_{9}(25) )}$

$:\implies\large\red{ \bf25}$

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$\maltese \large \: \sf {3}^{ \bigg( \dfrac{3}{ log_{ \sqrt{6} }(3) } \bigg)}$

$: \implies \large \sf{ {3}^{3 \times log_{3}( \sqrt{6} ) } }$

$: \implies\large \sf {3}^{ log_{3}( (\sqrt{6}) ^{3} ) } = { (\sqrt{6}) }^{3}$

$: \implies \large \red{\bf6 \sqrt{6} }$

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$\maltese \: \sf{(7)}^{ \bigg(\dfrac{2}{ log_{25}(7) } \bigg) }$

$: \implies\sf \large { { \sqrt{ 7}}^{2 \times log_{7}( 25 ) } }$

$: \implies \sf \large{7}^{ \frac{1}{2} \times 2 \times log_{7}(25) }$

$: \implies\red{ \bf 25}$

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$\maltese \: \large\sf{(125)}^{ ( log_{25}(6) ) }$

$: \implies \sf \large {5}^{3 \times log_{ {5}^{2} }(6) }$

$: \implies \sf \large 5 ^{ \frac{3}{2} \times log_{5}(6) }$

$: \implies \sf {6}^{ \frac{3}{2} }$

$:\implies\red{\bf 6 \sqrt{6}}$

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$\dag \: \underline \frak{Substituting \: the \: Above \: values \: in \: the \: given\: expression : - }$

$\rightarrow \sf \dfrac{25 + 6 \sqrt{6} }{409} \times 25 - 6 \sqrt{6}$

$\rightarrow \sf \dfrac{ \cancel{409}}{ \cancel{409}}$

$\rightarrow \boxed {\pink{{ \frak{1}}}} \bigstar$

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Hope u got what u were looking for :D