Question

Prove that log14/15+log28/27+log405/196=log2

Answer 1

SOLUTION

TO PROVE

$\displaystyle \sf{ log \bigg( \frac{14}{15} \bigg) + log \bigg( \frac{28}{27} \bigg) +log \bigg( \frac{405}{196} \bigg) = log(2) }$

FORMULA TO BE IMPLEMENTED

We are aware of the formula on logarithm that

$\sf{1. \: \: \: log( {a}^{n} ) = n log(a) }$

$\sf{2. \: \: log(ab) = log(a) + log(b) }$

$\displaystyle \sf{3. \: \: log \bigg( \frac{a}{b} \bigg) = log(a) - log(b) }$

$\sf{4. \: \: log_{a}(a) = 1}$

EVALUATION

LHS

$\displaystyle \sf{ = log \bigg( \frac{14}{15} \bigg) + log \bigg( \frac{28}{27} \bigg) +log \bigg( \frac{405}{196} \bigg) }$

$\displaystyle \sf{ = log \bigg( \frac{14}{15} \times \frac{28}{27} \bigg) +log \bigg( \frac{405}{196} \bigg) }$

$\displaystyle \sf{ = log \bigg( \frac{14}{15} \times \frac{28}{27} \times \frac{405}{196} \bigg) }$

$\displaystyle \sf{ = log \bigg( \frac{1}{15} \times \frac{28}{1} \times \frac{15}{14} \bigg) }$

$\displaystyle \sf{ = log \bigg( \frac{28}{15} \times \frac{15}{14} \bigg) }$

$\displaystyle \sf{ = log \bigg( \frac{2}{1}\bigg) }$

$\displaystyle \sf{ = log \: 2 }$

= RHS

Hence proved

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