Question

Express as single logarithm 5 log 25/24 - 3 log 80/81 - 7 log 15/10​

Answer 1

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

Given expression is

$\rm :\longmapsto\:5log\bigg(\dfrac{25}{24} \bigg) - 3log\bigg(\dfrac{80}{81} \bigg) - 7log\bigg(\dfrac{15}{10} \bigg)$

can be rewritten as

$\rm :\longmapsto\:5log\bigg(\dfrac{25}{24} \bigg) - 3log\bigg(\dfrac{80}{81} \bigg) - 7log\bigg(\dfrac{3}{2} \bigg)$

We know,

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

So, using this identity we get,

$\rm \: = \: \: {log\bigg(\dfrac{25}{24} \bigg)}^{5} + {log\bigg(\dfrac{80}{81} \bigg)}^{ - 3} + {log\bigg(\dfrac{3}{2} \bigg)}^{ - 7}$

$\rm \: = \: \: {log\bigg(\dfrac{25}{24} \bigg)}^{5} + {log\bigg(\dfrac{81}{80} \bigg)}^{3} + {log\bigg(\dfrac{2}{3} \bigg)}^{ 7}$

We know,

$\boxed{ \bf{ \: logx + logy = logxy}}$

$\rm \: = \: \: log \bigg \{ {\bigg(\dfrac{25}{24} \bigg)}^{5} \times {\bigg(\dfrac{81}{80} \bigg)}^{3} \times {\bigg(\dfrac{2}{3} \bigg)}^{7} \bigg \}$

$\rm \: = \: \: log \bigg \{ {\bigg(\dfrac{ {5}^{2} }{ {2}^{3} \times 3} \bigg)}^{5} \times {\bigg(\dfrac{ {3}^{4} }{ {2}^{4} \times 5} \bigg)}^{3} \times {\bigg(\dfrac{2}{3} \bigg)}^{7} \bigg \}$

$\rm \: = \: \: log \bigg \{ {\bigg(\dfrac{ {5}^{10} }{ {2}^{15} \times {3}^{5} } \bigg)}^{} \times {\bigg(\dfrac{ {3}^{12} }{ {2}^{12} \times {5}^{3} } \bigg)}^{} \times {\bigg(\dfrac{ {2}^{7} }{ {3}^{7} } \bigg)}^{} \bigg \}$

$\rm \: = \: \: log \bigg \{ {\bigg(\dfrac{ {5}^{7} }{ {2}^{20} } \bigg)}^{} \bigg \}$

Additional Information :-

$\boxed{ \bf{ \: log(x \div y) = logx - logy}}$

$\boxed{ \bf{ \: log_{x}(x) = 1}}$

$\boxed{ \bf{ \: log_{ {x}^{y} }( {x}^{z} ) = \frac{z}{y} }}$

$\boxed{ \bf{ \: {e}^{logx} = x}}$

$\boxed{ \bf{ \: {e}^{ylogx} = {x}^{y} }}$

$\boxed{ \bf{ \: {a}^{ log_{a}(x) } = x}}$