Question

$\\ \mathbf{QUESTION} : -$
The value of$\rm \: \dfrac{1}{ log_{2}n } + \dfrac{1 }{log_{3}n} + \dots + \dfrac{1}{ log_{43}n }$ is

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Answer 1

$\red{ \large \qquad \underline{\underline{ \pmb{\sf Some \: important \: formulae : }}}}$

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${ \color{blue}\qquad \dashrightarrow \: {\pmb{ \sf{ log \: _{a} \: (mn) = log \: _{a} \: m + log \: _{a} \: n}}}}$

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$\color{blue}{\qquad \dashrightarrow \: \pmb{ \sf{ \: log \: _{a} \dfrac{m}{n} = log \: _{a} \: m - log \: _{a} \: n }}}$

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$\color{blue}{{ \qquad \dashrightarrow \: \pmb{ \sf{ log \: _{a} \: {m}^{n} = n \: log \: _{a} \: m }} }}$

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$\color{blue}{ \qquad \dashrightarrow \: \pmb{ \sf{ log \: _{a} \: m = \dfrac{ log \: _{b} \: m }{ log \: _{b} \: a} }}}$

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$\color{blue}{ \qquad \dashrightarrow \: \pmb{ \sf{ log \: _{a} \: a =1 }}}$

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$\color{blue}{ \qquad \dashrightarrow \: \pmb{ \sf{ {a}^{ \: log \: _{a} \: m} = m }}}$

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$\color{blue}{ \qquad \dashrightarrow \: \pmb { \sf{ log \: _{a} \: 1 = 0}}}$

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Answer 2

Answer in the attachment.

Hope it helps you :)