Question

Find the value of log 400 base 2√5+log 0.0144 base 0.12+2^2log 5 base 2

Answer 1

SOLUTION

TO DETERMINE

The value of

$\displaystyle\sf{ log_{2 \sqrt{5} }(400) + log_{0.12}(0.0144) + {2}^{2 log_{2}(5) } }$

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

$\displaystyle\sf{ log_{2 \sqrt{5} }(400) + log_{0.12}(0.0144) + {2}^{2 log_{2}(5) } }$

$\displaystyle\sf{ = log_{2 \sqrt{5} }( {(2 \sqrt{5}) }^{4} ) + log_{0.12}( {(0.12)}^{2} ) + {2}^{ log_{2}( {5}^{2} ) } }$

$\displaystyle\sf{ = 4 log_{2 \sqrt{5} }{(2 \sqrt{5}) }^{} + 2 \: log_{0.12} {(0.12)}^{} + {5}^{2} }$

$\displaystyle\sf{ = 4 + 2 + 25}$

$\displaystyle\sf{ = 31}$

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