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Chapter 3: Problem 105

Evaluate each expression without using a calculator. $$\log _{3}\left(\log _{7} 7\right)$$

Short Answer

Expert verified
The solution to the expression \( \log_{3}(\log_{7}7) \) is 0.

Step by step solution

01

Understand the given expression

The given expression is \( \log_{3}(\log_{7}7) \). This is a log of a log. It may seem complex, but can be simplified using logarithm properties.
02

Simplify the inner logarithm

The expression inside the outer logarithmic function is \( \log_{7}7 \). According to the rule \( \log_b(b) = 1 \), this simplifies to 1. Hence, the expression becomes \( \log_{3}1 \).
03

Simplify the outer logarithm

Now simplify \( \log_{3} 1 \) using the same rule. Since anything to the power of 0 gives 1, \( \log_{3}1 = 0 \). This is our result.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Logarithm Properties
Logarithms have unique properties that simplify complex expressions. These properties are essential to tackle calculations where larger operations break down into more manageable steps.
Understanding them is crucial for evaluating expressions without technology. One basic property is that
  • \( \log_b(b) = 1 \)
This means the log of a number with its base is always 1. Another handy property includes:
  • \( \log_b(1) = 0 \)
This expresses that the log of 1, regardless of the base, equals zero.
These properties allow us to simplify complex logarithmic expressions effectively, as seen in the original exercise. Mastery of these rules ensures easier problem-solving overall.
Simplifying Logarithms
Simplifying logarithms involves applying core properties to break down expressions. The goal is to reduce them to simpler forms we can easily evaluate. For instance, if we have the expression \( \log_{3}(\log_{7} 7) \), the first step is tackling the inner logarithm.
This expression \( \log_{7}7 \) utilizes the property where \( \log_b(b) = 1 \). Therefore, the inner log simplifies directly to 1, as in \( \log_{7}7 = 1 \).
This greatly reduces the complexity of the expression to \( \log_{3}1 \). Understanding how to apply these principles lets us simplify logs without requiring calculators, saving time and effort.
Evaluating Expressions
Evaluating expressions, especially those involving logarithms, requires methodically applying proper rules. Consider the final result after simplification: \( \log_{3}1 \). Using the principle where \( \log_b(1) = 0 \), you can immediately see that this reduces to 0.
Why does this happen? Any number raised to the power 0 is 1, which reveals that \( \log_{3}1 \) indeed equals 0.
By mastering these evaluations, students develop the capability to perform quick assessments and validate results. With this understanding, tackling logarithmic expressions becomes less overwhelming and more intuitive, equipping learners with skills that span beyond mathematics.

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Most popular questions from this chapter

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