Question

log base 2[log base 2{log base 2(log 81 base 3)}]​

Answer 1

$\displaystyle \sf{ log_{2}[ log_{2} \{ log_{3}(81) \} ] } = \bf \: 1$

Correct question : Find the value of

$\displaystyle \sf{ log_{2}[ log_{2} \{ log_{3}(81) \} ] }$

Given :

$\displaystyle \sf{ log_{2}[ log_{2} \{ log_{3}(81) \} ] }$

To find :

The value of the expression

Formula :

We are aware of the formula on logarithm that

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

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

Solution :

Step 1 of 2 :

Write down the given expression

Here the given expression is

$\displaystyle \sf{ log_{2}[ log_{2} \{ log_{3}(81) \} ] }$

Step 2 of 2 :

Find the value of the expression

$\displaystyle \sf{ log_{2}[ log_{2} \{ log_{3}(81) \} ] }$

$\displaystyle \sf{ = log_{2}[ log_{2} \{ log_{3}( {3}^{4} ) \} ] }$

$\displaystyle \sf{ = log_{2}[ log_{2} \{ 4 log_{3}( {3}^{} ) \} ] }\: \: \: \bigg[ \: \because \:log( {a}^{n} ) = n log(a) \bigg]$

$\displaystyle \sf{ = log_{2}[ log_{2}(4 \times 1 ) ] }\: \: \: \bigg[ \: \because \: log_{a}(a) = 1\bigg]$

$\displaystyle \sf{ = log_{2}[ log_{2}(4 ) ] }$

$\displaystyle \sf{ = log_{2}[ log_{2}( {2}^{2} ) ] }$

$\displaystyle \sf{ = log_{2}[2 log_{2}(2 ) ] }\: \: \: \bigg[ \: \because \: log( {a}^{n} ) = n log(a)\bigg]$

$\displaystyle \sf{ = log_{2}[2 \times 1 ] }\: \: \: \bigg[ \: \because \: log_{a}(a) = 1 \bigg]$

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

$\displaystyle \sf{ = 1 }\: \: \: \bigg[ \: \because \: log_{a}(a) = 1 \bigg]$

$\displaystyle \sf{ \therefore \: \: log_{2}[ log_{2} \{ log_{3}(81) \} ] } = \: 1$

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Answer 2

Answer:

 The value of $log_{2}(log_{2}(log_{2 }(log_{3}81)))$ is  0.

Step-by-step explanation:

logarithm formulas:

• Some of the logarithm formulas are given below  

           $loga^{n}=nloga$

           $log_{a} {a = 1$

•  Now apply above formulas to $log_{2}(log_{2}(log_{2 }(log_{3}81)))$  

               $log_{2}(log_{2}(log_{2 }(log_{3}81)))$

           $=log_{2}(log_{2}(log_{2 }(log_{3}3^{4} )))$

           $=log_{2}(log_{2}(log_{2 }(4log_{3}3 )))$

           $=log_{2}(log_{2}(log_{2 }4 ))$

           $=log_{2}(log_{2}(log_{2 }2^{2} ))$

           $=log_{2}(log_{2}(2log_{2 }2 ))$

           $=log_{2}(log_{2}2 )$

           $=log_{2}1$

           $=0$

Hence , $log_{2}(log_{2}(log_{2 }(log_{3}81)))=0$

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