Question
Answers

Find the value of t for the given expression.
 ${{\log }_{2}}\left[ {{\log }_{3}}\left( \log 2t \right) \right]=1$
A. 512
B. 128
C. 1024
D. None

Answer
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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.