Question

evaluate log base 2 log base 3 log base 2 512

Answer 1

the answer is one.
According to the question
$log_{2}( log_{3}(log_{2}(512) ) )$
Also

$512 = {2}^{9}$
so now it becomes,
$log_{2}( log_{3}(9) )$
we know,
$9 = {3}^{2}$
now the whole equation reduces to,
$log_{2}(2) )$
which becomes 1

Answer 2

$\sf log_{2}( log_{3}( log_{2}(512) ) ) = 1$

Given : $\sf log_{2}( log_{3}( log_{2}(512) ) )$

To find : The value

Tip :

Formula to be used

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

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

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

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

Solution :

Step 1 of 2 :

Write down the given expression

The given expression is

$\sf log_{2}( log_{3}( log_{2}(512) ) )$

Step 2 of 2 :

Simplify the given expression

We simplify the given expression as below

$\sf log_{2}( log_{3}( log_{2}(512) ) )$

$\sf = log_{2}( log_{3}( log_{2}( {2}^{9} ) ) )$

$\sf = log_{2}( log_{3}(9 log_{2}( {2}^{} ) ) )$

$\sf = log_{2}( log_{3}(9 \times 1 ) )$

$\sf = log_{2}( log_{3}(9 ) )$

$\sf = log_{2}( log_{3}( {3}^{2} ) )$

$\sf = log_{2}( 2log_{3}( {3}^{} ) )$

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

$\sf = 1$

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