Question:medium

What is the value of \(\log\log\sqrt{\sqrt[5]{\sqrt[10]{\sqrt[10]{10}}}}\) ?

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Turn every nested root into a fractional exponent and multiply the exponents together before taking the two logs.
Updated On: Jul 21, 2026
  • -3
  • -2
  • -1
  • 1
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The Correct Option is A

Solution and Explanation

Step 1: Peel the roots off one at a time from the inside out.
Let \(x_1=\sqrt[10]{10}\), so \(\log x_1 = \dfrac{1}{10}\log 10 = \dfrac{1}{10}\).

Step 2: Take the next tenth root.
Let \(x_2=\sqrt[10]{x_1}\). Using the power rule for logs, \(\log x_2 = \dfrac{1}{10}\log x_1 = \dfrac{1}{10}\times\dfrac{1}{10}=\dfrac{1}{100}\).

Step 3: Take the fifth root.
Let \(x_3=\sqrt[5]{x_2}\). Then \(\log x_3=\dfrac15\log x_2=\dfrac15\times\dfrac{1}{100}=\dfrac{1}{500}\).

Step 4: Take the outer square root.
Let \(x_4=\sqrt{x_3}\). Then \(\log x_4=\dfrac12\log x_3=\dfrac12\times\dfrac{1}{500}=\dfrac{1}{1000}\).
Notice this already matches the first log we need, since \(x_4\) is exactly the number inside the outer log log.

Step 5: Take the final log.
We need \(\log(\log x_4) = \log\left(\dfrac{1}{1000}\right)\). Since \(\dfrac{1}{1000}=10^{-3}\), this equals \(-3\).
Building the answer step by step through each root confirms the same result as multiplying the exponents directly. \[ \boxed{-3} \]
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