Question

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

• A. -3
• B. -2
• C. -1
• D. 1

Hint:

Turn every nested root into a fractional exponent and multiply the exponents together before taking the two logs.

Answer 1

The Correct Option is A

Step 1: Rewrite every root as a fractional power.
A radical is just a power with a fractional exponent, so \(\sqrt[n]{x}=x^{1/n}\). Reading the expression from the innermost root outward, we have a tenth root, then another tenth root, then a fifth root, then a square root, all applied to 10.

Step 2: Combine the exponents into one.
Taking a tenth root, then a tenth root again, then a fifth root, then a square root multiplies the exponents together:
\[ \left(\left(\left(10^{1/10}\right)^{1/10}\right)^{1/5}\right)^{1/2} = 10^{\frac{1}{10}\times\frac{1}{10}\times\frac15\times\frac12} \]
\[ \frac{1}{10}\times\frac{1}{10}\times\frac15\times\frac12 = \frac{1}{1000} \]
So the whole nested radical equals \(10^{1/1000}\).

Step 3: Apply the first log.
Using \(\log(10^k)=k\) (log to base 10), \[ \log\left(10^{1/1000}\right) = \frac{1}{1000} = 0.001 \]

Step 4: Apply the second log.
Now we need \(\log(0.001)\). Since \(0.001 = 10^{-3}\),
\[ \log(0.001) = \log(10^{-3}) = -3 \]

Final Answer:
The value of the expression is \(-3\). \[ \boxed{-3} \]