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Chapter 10: Problem 86

Evaluate each logarithm. $$ \log _{100} 100 $$

Short Answer

Expert verified
The evaluation of the logarithm \( \log _{100} 100 \) is 1.

Step by step solution

01

Identify the base and the argument

From \(\log _{100} 100\), the base is 100 and the argument is also 100.
02

Apply the rule of logarithm

A rule of logarithms states that \(\log_b(b) = 1\). So, when the base and the argument of a log function are the same, the output is 1.

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Key Concepts

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

Base of Logarithm
In the world of logarithms, understanding the base is crucial. The base of a logarithm is the number you're raising to a power to result in the argument. In simple terms, if you think of a logarithm \( \log_b(a) \), the small number \( b \) placed lower than the main text, is the base.
For example, in the logarithm \( \log_{100}(100) \), the base is 100. It's the foundation that tells us how many times we multiply it by itself to get the argument.
Different bases are used for different purposes.
  • The most common base is 10, known as the common logarithm.
  • Base \( e \) (approximately 2.718) is used for natural logarithms, denoted as \( \ln \).
  • Understanding the role of the base helps in simplifying logarithms and solving real-world problems.
Argument of Logarithm
The argument of a logarithm is the number we're interested in. It's what results when the base has been raised to the calculated power. Look at \( \log_b(a) \); here, \( a \) is the argument.
In the exercise \( \log_{100}(100)\), 100 is our argument.
This simply means you're exploring what power you need to raise the base 100 to, in order to get back 100.
Identifying arguments correctly can help easily solve log problems and understand complex equations.
A few tips to remember about arguments:
  • Always is a positive number, as logs of negative numbers or zero are undefined.
  • Greater emphasis should be placed on correctly identifying them as it impacts the final solution.
  • Sometimes, the argument itself might give clues to shortcuts, like in cases where it's the same as the base.
Properties of Logarithms
Logarithms come with their own set of properties or rules that help simplify and solve equations. These rules are universal and help you handle complex expressions by breaking them down. Let's explore some key properties:
1. **Logarithm of the base equals one:**
  • This property states \( \log_b(b) = 1 \). Essentially, for any logarithm, if the base and the argument are the same, the logarithm simplifies to 1. This is why \( \log_{100}(100) = 1 \).
2. **Product rule:**
  • Says \( \log_b(xy) = \log_b(x) + \log_b(y) \). Use this to break down logs into simpler parts when adjacent.
3. **Quotient rule:**
  • Indicates \( \log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y) \), helping you solve divisions within logs.
4. **Power rule:**
  • Explains \( \log_b(x^p) = p \cdot \log_b(x) \), which means that the exponent of the argument can be taken out and multiplied.
These properties make working with logarithms intuitive and efficient. Use them to simplify equations, making calculations more manageable. Understanding these can aid in tackling both academic problems and practical applications involving exponential growth, sound intensity, and more.

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