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

Ch. 4 - Exponential and Logarithmic Functions

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

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Blitzer 8th Edition

Ch. 4 - Exponential and Logarithmic Functions

Problem 144

Chapter 5, Problem 144

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

Verified step by step guidance

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 \( \log_2 8 = 3 \).

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 \( \log_3 3 = 1 \).

Now, substitute this result into the outermost logarithm: \( \log_4 [\log_3 (\log_2 8)] = \log_4 1 \).

Recall that for any base \( b > 0 \) and \( b \neq 1 \), \( \log_b 1 = 0 \) because \( b^0 = 1 \). Therefore, \( \log_4 1 = 0 \).

Thus, the exact value of the original expression \( \log_4 [\log_3 (\log_2 8)] \) is \( 0 \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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Change of Base Property

Related Practice

Textbook Question

Without using a calculator, find the exact value of: [log₃ 81 - log_𝝅 1]/[log_2√2 8 - log 0.001]

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Textbook Question

Exercises 137–139 will help you prepare for the material covered in the next section. Solve: (x + 2)/(4x + 3) = 1/x

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Textbook Question

145. Without using a calculator, determine which is the greater number: log₄ 60 or log₃ 40.

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