Question

If log a (log 3 (log 2 (512))) = 1 then a = ​

Answer 1

Answer:

Ans: 2

solution:

log a( log 3(log 2(512)))=1

log a( log 3(log 2(2^9)))=1 2^9 = 512

log a( log 3( 9 log 2(2)))=1 log m^n = n log m

log a( log 3( 9 *1 ))=1 log m m= 1

log a( log 3 ( 9 )) =1

log a( log 3 (3^2)) =1

log a( 2 log 3 3)=1

log a(2*(1))=1

log a (2)=1

a= 2

Answer 2

$\sf \: If \: log_{a}( log_{3}( log_{2}(512) ) ) = 1 \: \: then \: a = \bf 2$

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

To find : The value of a

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 equation

The given equation is

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

Step 2 of 2 :

Simplify the given expression

We simplify the given expression as below

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

$\sf \implies log_{a}( log_{3}( log_{2}( {2}^{9} ) ) ) =1$

$\sf \implies log_{a}( log_{3}(9 log_{2}( {2}^{} ) ) ) =1$

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

$\sf \implies log_{a}( log_{3}(9 ) )=1$

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

$\sf \implies log_{a}( 2log_{3}( {3}^{} ) )=1$

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

$\sf \implies a=2$

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