Without using a calculator, find the exact value of log₄ [log₃ (log₂ 8)].

Verified step by step guidance

1. Start by evaluating the innermost logarithm: \( \log_2 8 \). Recall that \( \log_b a = c \) means \( b^c = a \). Since \( 2^3 = 8 \), we have \( \log_2 8 = 3 \).
2. Next, substitute this value into the next logarithm: \( \log_3 (\log_2 8) = \log_3 3 \). Using the same definition, since \( 3^1 = 3 \), it follows that \( \log_3 3 = 1 \).
3. Now, substitute this result into the outermost logarithm: \( \log_4 [\log_3 (\log_2 8)] = \log_4 1 \).
4. Recall that for any base \( b > 0 \) and \( b \neq 1 \), \( \log_b 1 = 0 \) because \( b^0 = 1 \). Therefore, \( \log_4 1 = 0 \).
5. Thus, the exact value of the original expression \( \log_4 [\log_3 (\log_2 8)] \) is \( 0 \).

Key Concepts

Change of Base and Nested Logarithms

Understanding how to evaluate nested logarithms requires recognizing the order of operations and simplifying from the innermost logarithm outward. Each logarithm must be evaluated exactly before applying the next, ensuring clarity in the base and argument at each step.

Evaluating Logarithms with Simple Arguments

Logarithms with arguments that are powers of the base can be simplified using the identity log_b(b^k) = k. For example, log₂ 8 simplifies to 3 because 8 = 2^3. This simplification is key to finding exact values without a calculator.

Properties of Logarithms and Exact Values

Knowing logarithm properties, such as log_b(1) = 0 and log_b(b) = 1, helps in simplifying expressions. Exact values are found by expressing numbers as powers of the base and applying these properties step-by-step to avoid approximations.