Problem 144 - College Algebra Essentials

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 log

Let's simplify the innermost part of the expression first, that is \( \log_2 8 \). It asks the question: '2 to the power of what number gives us 8?' Since \( 2^3 = 8 \), then \( \log_2 8 = 3 \). The original equation simplifies to \( \log_4 [\log_3 3] \).

Step 2: Simplify the second log

Again, similar to Step 1, simplification of \( \log_3 3 \) will be done. The log asks '3 to the power of what number gives 3?' Since \( 3^1 = 3 \), then \( \log_3 3 = 1 \). This leads to further simplification of the equation, which is now \( \log_4 1 \).

Step 3: Simplify the final log

The simplified equation \( \log_4 1 \) asks '4 to the power of what number gives 1?' Since any number raised to the power of 0 always gives 1, the result would be \( \log_4 1 = 0 \).