Question

Find the value of :-

$\\ \boxed{ \sf log_{3}(2). log_{4}(3) . log_{5}(4) ... log_{11}(10) . log_{2}(11) }$

Topic - Logarithm ​

Answer 1

$\maltese \: \large \underline{ \pmb {\orange{{ \frak{ Base\: Changing \: Property : - }}}}}$

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

$\maltese \: \large \underline{ \pmb {\orange{{ \frak{ Required \: Solution : - }}}}}$

The first step is to apply the base changing property in all the individual Logarithmic expression.

$\\ : \implies \sf \frac{ log(2) }{ log(3) } \times \frac{ log(3) }{ log(4) } \times \frac{ log(4) }{ log(5) } . \: . \: . \: . \: . \: \frac{ log(10) }{ log(11) } \times \frac{ log(11) }{ log(2) }$

• Observe the pattern, we have got the reciprocal of each log in its consecutive multiple.

$\\ : \implies \sf \dfrac{ log(2) }{ log(2) }$

$\\ : \implies {\boxed{ \pink{{\sf 1}}}} \bigstar$

hence, our required Answer is 1.

Answer 2

$\underline{\purple{\ddot{\MasterRohith}}}$

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