Question

log 216√6 to the base√6 = ?​

Answer 1

Answer:

I am not sure if you want your answer

Answer 2

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

We know that, .

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

Now,

Given that

$\rm :\longmapsto\: log_{ \sqrt{6} }(216 \sqrt{6} )$

Consider,

$\rm \: = \: \: log_{ \sqrt{6} }(6 \times 6 \times 6 \times \sqrt{6} )$

$\rm \: = \: \: log_{ \sqrt{6} }( {( \sqrt{6} )}^{2} \times {( \sqrt{6} )}^{2} \times {( \sqrt{6}) }^{2} \times \sqrt{6} )$

$\rm \: = \: \: log_{ \sqrt{6} }( {( \sqrt{6} )}^{2 + 2 + 2 + 1} )$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \red{\bigg \{ \because \: {a}^{x} \times {a}^{y} = {a}^{x + y} \bigg \}}$

$\rm \: = \: \: log_{ \sqrt{6} }( {( \sqrt{6} )}^{7} )$

$\rm \: = \: \:7$

Hence,

$\bf :\longmapsto\: log_{ \sqrt{6} }(216 \sqrt{6} ) = 7$

Additional Information :-

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

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

$\rm :\longmapsto\: {e}^{logx} = x$

$\rm :\longmapsto\: {e}^{ylogx} = {x}^{y}$

$\rm :\longmapsto\: {a}^{ log_{a}(x) } = x$

$\rm :\longmapsto\: {a}^{ ylog_{a}(x) } = {x}^{y}$