Source: Extra Practice

Evaluate the value of the nested logarithmic expression: log⁡2(log⁡2(log⁡3(log⁡3273)))\log_{2}\left(\log_{2}\left(\log_{3}\left(\log_{3} 27^3\right)\right)\right).

Options

Option A is correct

00

Option B

11

Option C

22

Option D

33

Explanation

We simplify the expression step-by-step from the innermost logarithm outward:

  1. First, simplify the innermost argument: 273=(33)3=3927^3 = (3^3)^3 = 3^9.
  2. Evaluate the innermost logarithm: log⁡3(39)=9log⁡3(3)=9\log_{3}(3^9) = 9 \log_{3}(3) = 9.
  3. Substitute this value back into the next layer: log⁡3(9)=log⁡3(32)=2\log_{3}(9) = \log_{3}(3^2) = 2.
  4. Substitute this value into the next layer: log⁡2(2)=1\log_{2}(2) = 1.
  5. Finally, evaluate the outermost logarithm: log⁡2(1)=0\log_{2}(1) = 0.

Thus, the final value is 00. The other options represent incomplete evaluations or algebra errors.