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${{\log }_{2}}\left[ {{\log }_{3}}\left( \log 2t \right) \right]=1$

A. 512

B. 128

C. 1024

D. None

Answer

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148.2k+ views

Hint: Use logarithm properties and formulas to simplify the equation.Start by opening the outermost log by the formula ${{\log }_{a}}b=x,b={{a}^{x}}$ to find the value of t.

“Complete step-by-step answer:”

${{\log }_{2}}\left[ {{\log }_{3}}\left( \log 2t \right) \right]=1$

Base of the above logarithm is 2.

We write the above log in exponential form.

$\left[ When\ {{\log }_{a}}b=x,b={{a}^{x}} \right]$

Therefore, ${{\log }_{3}}\left( {{\log }_{2}}t \right)={{2}^{1}}$

$\Rightarrow {{\log }_{3}}\left( {{\log }_{2}}t \right)=2$

Base of the above logarithm is 3.

We again write it in exponential form.

Therefore, ${{\log }_{2}}t={{3}^{2}}$

$\Rightarrow {{\log }_{2}}t=9$

The base of the above logarithm is 2.

We write it in exponential form.

Therefore, $t={{2}^{9}}$

$\Rightarrow t=512$

Note: While solving the above question, do the calculation carefully.

Start by opening the outermost log and apply the logarithmic properties and formulas wherever necessary.

“Complete step-by-step answer:”

${{\log }_{2}}\left[ {{\log }_{3}}\left( \log 2t \right) \right]=1$

Base of the above logarithm is 2.

We write the above log in exponential form.

$\left[ When\ {{\log }_{a}}b=x,b={{a}^{x}} \right]$

Therefore, ${{\log }_{3}}\left( {{\log }_{2}}t \right)={{2}^{1}}$

$\Rightarrow {{\log }_{3}}\left( {{\log }_{2}}t \right)=2$

Base of the above logarithm is 3.

We again write it in exponential form.

Therefore, ${{\log }_{2}}t={{3}^{2}}$

$\Rightarrow {{\log }_{2}}t=9$

The base of the above logarithm is 2.

We write it in exponential form.

Therefore, $t={{2}^{9}}$

$\Rightarrow t=512$

Note: While solving the above question, do the calculation carefully.

Start by opening the outermost log and apply the logarithmic properties and formulas wherever necessary.