Textbook Question
In Exercises 105–108, evaluate each expression without using a calculator. log (ln e)
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6. Exponential & Logarithmic Functions
Introduction to Logarithms
3:49 minutesProblem 144
Textbook Question
Without using a calculator, find the exact value of log4 [log 3 (log₂ 8)].
Verified step by step guidance1
insert step 1> Start by evaluating the innermost logarithm: \( \log_2 8 \). Since 8 is a power of 2, express 8 as \( 2^3 \).
insert step 2> Use the property of logarithms that \( \log_b (b^x) = x \) to find \( \log_2 8 = \log_2 (2^3) = 3 \).
insert step 3> Substitute the result from step 2 into the next logarithm: \( \log_3 (\log_2 8) = \log_3 3 \).
insert step 4> Again, use the property of logarithms that \( \log_b (b^x) = x \) to find \( \log_3 3 = 1 \).
insert step 5> Substitute the result from step 4 into the outermost logarithm: \( \log_4 (\log_3 3) = \log_4 1 \). Since any logarithm of 1 is 0, \( \log_4 1 = 0 \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Logarithmic Functions
Logarithmic functions are the inverses of exponential functions. The logarithm log_b(a) answers the question: 'To what power must the base b be raised to obtain a?' Understanding how to manipulate and evaluate logarithmic expressions is crucial for solving problems involving logs.
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Graphs of Logarithmic Functions
Change of Base Formula
The change of base formula allows you to convert logarithms from one base to another, expressed as log_b(a) = log_k(a) / log_k(b) for any positive k. This is particularly useful when dealing with logarithms of different bases, enabling easier calculations and simplifications.
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Properties of Logarithms
Properties of logarithms, such as the product, quotient, and power rules, provide essential tools for simplifying logarithmic expressions. For example, log_b(mn) = log_b(m) + log_b(n) and log_b(m/n) = log_b(m) - log_b(n). Mastery of these properties is vital for breaking down complex logarithmic problems.
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