Question

don't post irrelevant answers
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Answer 1

kindly check the attachment for the stepwise solution and answer.

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Answer 2

$\displaystyle\huge\red{\underline{\underline{Solution}}}$

FORMULA TO BE IMPLEMENTED

1.

$log_{x}(a) + log_{x}(b) = log_{x}(ab)$

2.

$log_{x}( {y}^{m} ) = m log_{x}(y)$

3.

$1 + 2 + 3 + .... + n = \frac{n(n + 1)}{2}$

CALCULATION

1.

$2logx \: + 2log {x}^{2} + 2log {x}^{3} + ........ + 2log {x}^{n}$

$= 2logx \: + (2 \times 2)log {x} + (2 \times 3)log {x}+ ........ + (2 \times n)log {x}$

$= 2logx \: + 4log {x} + 6log {x}+ ........ + 2nlog {x}$

$= 2logx \: (1 + 2 + 3..... + n)$

$= \displaystyle \: 2logx \: \times \frac{n(n + 1)}{2}$

$= n(n + 1)logx$

2.

$\displaystyle \: log_{2}(1 + \frac{1}{2} ) + log_{2}(1 + \frac{1}{3} ) + log_{2}(1 + \frac{1}{4} ) + .... + log_{2}(1 + \frac{1}{31} )$

$= \displaystyle \: log_{2}( \frac{3}{2} ) + log_{2}(\frac{4}{3} ) + log_{2}(\frac{5}{4} ) + .... + log_{2}(\frac{32}{31} )$

$= \displaystyle \: log_{2}( \frac{3}{2} \times \frac{4}{3} \times \frac{5}{4} \times ..... \times \frac{32}{31} )$

$= \displaystyle \: log_{2}( \frac{32}{2} )$

$= \displaystyle \: log_{2}(16)$

$= \displaystyle \: log_{2}( {2}^{4} )$

$= 4 log_{2}(2)$

$= 4 \times 1$

$= 4$