Question

If log X = (loga Y/loga X)^k , then k =

Answer 1

SOLUTION

GIVEN

$\displaystyle \sf{ \log x = { \bigg( \frac{ log_{a}y }{ log_{a}x } \bigg)}^{k} }$

TO DETERMINE

The value of k

EVALUATION

Here it is given that

$\displaystyle \sf{ \log x = { \bigg( \frac{ log_{a}y }{ log_{a}x } \bigg)}^{k} }$

Taking logarithm in both sides we get

$\displaystyle \sf{ \implies \: \log( \log x) = \log { \bigg( \frac{ \frac{ \log y}{ \log a} }{ \frac{ \log x}{ \log a } } \bigg)}^{k} }$

$\displaystyle \sf{ \implies \: \log( \log x) =k \log { \bigg( \frac{ \log y}{ \log x} \bigg)}}$

$\displaystyle \sf{ \implies \: \log( \log x) = k \log { \big( log_{x}y \big)}}$

$\displaystyle \sf{ \implies \:k = \frac{\log( \log x)}{\log { \big( log_{x}y \big)}} }$

FINAL ANSWER

$\boxed{ \: \: \displaystyle \sf{ \:k = \frac{\log( \log x)}{\log { \big( log_{x}y \big)}} } \: \: }$

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