Question

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

• A. -3
• B. -2
• C. -1
• D. 1
• E. 3

Hint:

Multiply all the root indices together: \(2\times5\times10\times10=1000\), so the whole radical equals \(10^{1/1000}\); then take log twice.

Answer 1

The Correct Option is A

Step 1: Simplify the innermost radicals first.
Start from the inside: \(\sqrt[10]{10}=10^{1/10}\).

Step 2: Apply the next tenth root.
\(\sqrt[10]{10^{1/10}}=\left(10^{1/10}\right)^{1/10}=10^{1/100}\).

Step 3: Apply the fifth root.
\(\sqrt[5]{10^{1/100}}=\left(10^{1/100}\right)^{1/5}=10^{1/500}\).

Step 4: Apply the outer square root.
\(\sqrt{10^{1/500}}=\left(10^{1/500}\right)^{1/2}=10^{1/1000}\).

Step 5: Take the first log.
\(\log\left(10^{1/1000}\right)=\dfrac{1}{1000}\times\log(10)=\dfrac{1}{1000}\) (using base-10 log, \(\log 10=1\)).

Step 6: Take the second (outer) log.
\(\log\left(\dfrac{1}{1000}\right)=\log(10^{-3})=-3\).

Step 7: Conclusion.
The value of the expression is -3.