Problem 46 - Algebra 2

Question

REASONING Use the properties of Logarithms to prove that \(\log _{a} \frac{1}{x}=-\log _{a} x\)

Step-by-Step Solution

Answer

The property \(\log_a \frac{1}{x} = -\log_a x\) is proved using the logarithm properties.

Step 1: Understand the Logarithm of a Fraction

We know from the properties of logarithms that the logarithm of a fraction can be expressed as the difference of the logarithms: \(\log_a \frac{1}{x} = \log_a (1) - \log_a (x)\).

Step 2: Apply the Property of Logarithm of 1

Recognize that the logarithm of 1 in any base is always 0: \(\log_a (1) = 0\). Therefore, the expression becomes \(\log_a \frac{1}{x} = 0 - \log_a (x)\).

Step 3: Simplify the Expression

Simplify the expression from the previous step: \(0 - \log_a (x) = -\log_a (x)\).

Step 4: Conclusion

We have shown that \(\log_a \frac{1}{x} = -\log_a (x)\) using the properties of logarithms.

Key Concepts

Logarithm of a Fraction

The concept of the logarithm of a fraction helps us simplify expressions where a fraction is involved inside a logarithm. To understand this, let's recall one of the fundamental properties of logarithms: the logarithm of a fraction is equal to the difference of the logarithms. This can be expressed as follows: \[ \log_a \frac{u}{v} = \log_a (u) - \log_a (v) \]In the exercise, we see this property applied to \[ \log_a \frac{1}{x} \]which gives us the expression:\[ \log_a (1) - \log_a (x) \]This step shows that fractions can be transformed into simpler expressions using subtraction. This property is very useful for simplifying complex logarithmic equations and is one of the key tools in problem-solving with logarithms.

Logarithm of 1

Now, let's explore the concept of the logarithm of 1. A key property of logarithms is that the logarithm of 1, regardless of the base, is always zero.This can be represented as:\[ \log_a (1) = 0 \]The reasoning behind this is straightforward. Since any number raised to the power of 0 equals 1 (e.g., \(a^0 = 1\)), it follows naturally that the logarithm of 1 should equal zero.In our problem, knowing that \(\log_a (1) = 0\) helps simplify the expression \[ \log_a \frac{1}{x} = \log_a (1) - \log_a (x) \]to\[ 0 - \log_a (x) \].This is a crucial step that makes it possible to simplify the expression entirely.

Logarithmic Identities

Logarithmic identities are rules that relate logarithms to one another, making it easier to simplify and manipulate mathematical expressions involving logs.Fundamental logarithmic identities include:

• Product Identity: \( \log_a (uv) = \log_a (u) + \log_a (v) \)
• Quotient Identity: \( \log_a \frac{u}{v} = \log_a (u) - \log_a (v) \)
• Power Identity: \( \log_a (u^n) = n \cdot \log_a (u) \)

These identities are the building blocks for solving complex logarithmic expressions and problems. In the exercise, we used the quotient identity to transform the log of a fraction into the difference of two logs.Mastering these identities is crucial for algebra and higher-level mathematics, as they allow for the decomposition, simplification, and resolution of expressions with logarithms.

Mathematical Proofs

Mathematical proofs are a process of demonstrating the truth of a statement using accepted mathematical principles and logic. Proofs can be simple or quite complex, depending on the topic.In the context of logarithms, they help establish the validity of properties and identities, like demonstrating that\[ \log_a \frac{1}{x} = -\log_a (x) \]A typical proof steps through each logical conclusion systematically:

• Start with known identities or properties.
• Apply these step-by-step to the given expression.
• Ensure each step logically follows from the previous one.
• Reach a conclusion that verifies the statement in question.

In our solved problem, we used mathematical proof to validate the relationship between the log of a fraction and the negation of a log, using known logarithmic properties. Such proofs reinforce the reliability and consistency of mathematical concepts and help build a deeper understanding of the subject.