Question

“The logarithms of the same number of different bases are different”. Justify your answer with an example.​

Answer 1

SOLUTION

TO JUSTIFY

“The logarithms of the same number of different bases are different”

FORMULA TO BE IMPLEMENTED

We are aware of the formula on logarithm that

$\sf{ 1. \: \: log_{x}( {y}^{n} ) = n log_{x}(y) }$

$\sf{ 2. \: \: log_{x}( x ) =1 }$

EVALUATION

We consider logarithm of 16 with two different bases 2 and 4

Now

$\sf{ log_{2}(16) }$

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

$\sf{ = 4 log_{2}( 2 ) } \: \: \: \: (by \: formula \: 1)$

$\sf{ = 4 \times 1 \: \: \: \: \:( by \: formula \: 2)}$

$= 4$

Again

$\sf{ log_{4}( 16) }$

$\sf{ = log_{4}( {4}^{2} ) }$

$\sf{ = 2 log_{4}(4) \: \: \: \: \:( by \: formula \: 1)}$

$\sf{ = 2 \times 1 \: \: \: \: \:( by \: formula \: 2)}$

$= 2$

Hence

$\sf{ log_{2}( 16) \ne \: log_{4}(16) }$

Hence The logarithms of the same number of different bases are different

Hence justified

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