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Chapter 9: Problem 23

Simplify. $$ \log _{7} 7 $$

Short Answer

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Step by step solution

01

Understand the Logarithmic Identity

The base-\(b\) logarithm of \(b\) itself, written as \(\log_b b\), is always equal to 1, because any number raised to the power of 1 is itself.
02

Apply the Identity to the Given Expression

In the expression \(\log_7 7\), both the base and the argument are 7. According to the logarithmic identity \(\log_b b = 1\), we can simplify \(\log_7 7\) as follows: \(\log_7 7 = 1\).

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Key Concepts

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

logarithmic identity
Logarithmic identities are foundational tools in understanding logarithms. One such identity is the fact that the base-\b logarithm of the base itself is always equal to 1. This is represented as \(\log_b b = 1\). To break this down, we can recall the definition of a logarithm: \( \log_b b = 1\) implies that \(b \) raised to the power of \(1 \) gives us \(b\). This is true for any positive number b. This identity simplifies many logarithmic expressions and is particularly useful for those beginning to learn about logarithms.
base and argument
In a logarithmic expression, the base and argument are crucial components. The base is the number that raises to a power to yield the argument. For example, in \( \log_7 7 \), \(7\) is both the base and argument. Understanding these elements of a logarithm is vital for manipulating and simplifying logarithmic expressions. When the base and argument are the same, like in \(\log_7 7\), it directly invokes the logarithmic identity \( \log_7 7 = 1 \). This relationship between the base and argument is a cornerstone of logarithmic properties and makes comprehension and simplification easier.
simplifying logarithmic expressions
Simplifying logarithmic expressions often requires using logarithmic identities and understanding the role of bases and arguments. For instance, to simplify \(\frac{\log_b (m \cdot n)}{\log_b (m) + \log_b (n)}\), you apply properties of logarithms. In the provided example, \( \log_7 7 \), the first step is identifying that the base and argument are the same. Using the logarithmic identity \( \log_b b = 1\), simplifies the expression instantaneously to 1. Such techniques rely heavily on recognizing patterns and repetitively applying rules. It鈥檚 about replacing complex-looking expressions with simpler, more familiar ones to reach an answer quickly and accurately.

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