Problem 144 - Algebra and Trigonometry

Question

Without using a calculator, find the exact value of \(\log _{4}\left[\log _{3}\left(\log _{2} 8\right)\right]\)

Step-by-Step Solution

Answer

The exact value of the given logarithmic expression is \(0\).

Step 1: Evaluating \( \log_{2}(8) \)

We start by evaluating the innermost logarithm \(\log_{2}(8)\). Since \(2^{3} = 8\), we know that \(\log_{2}(8) = 3\). Thus the original expression becomes \(\log_{4}[\log_{3}(3)]\).

Step 2: Evaluating \( \log_{3}(3) \)

Next we evaluate \(\log_{3}(3)\). Since \(3^{1} = 3\), we know that \(\log_{3}(3) = 1\). Therefore, the original expression now simplifies to \(\log_{4}(1)\).

Step 3: Evaluating \( \log_{4}(1) \)

Finally, we evaluate \(\log_{4}(1)\). Since \(4^{0} = 1\), we know that \(\log_{4}(1) = 0\). Hence, the original expression simplifies to 0 as its ultimate value.