Problem 144 - College Algebra

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 is 0.

Step 1: Simplify the Innermost Logarithm

Start with the innermost logarithm \( \log _{2} 8 \). We know that 2 raised to the power of 3 is 8. So, \( \log _{2} 8 = 3 \). Then we substitute this value back into the whole equation, giving \( \log _{4}\left[\log _{3}\left(3\right)\right] \).

Step 2: Simplify the Second Logarithm

Now we simplify the second logarithm \( \log _{3} 3 \). We know that 3 raised to the power of 1 is 3. So, \( \log _{3} 3 = 1 \). Then we substitute this value back into the whole equation, giving \( \log _{4} 1 \).

Step 3: Simplify the Final Logarithm

Now we simplify the final logarithm \( \log _{4} 1 \). We know that 4 raised to the power of 0 is 1. So, \( \log _{4} 1 = 0 \).