It looks like you're asking about a logarithmic expression. Let's clarify the expression:

You have:

\[
\log_2 \left( \frac{x^2 - y^2}{xy} \right)
\]

To simplify or rewrite this expression, we can break it down step-by-step.

### Step 1: Recognize the Structure of the Argument
The argument of the logarithm is:

\[
\frac{x^2 - y^2}{xy}
\]

We can recognize that \(x^2 - y^2\) is a difference of squares. This can be factored using the formula:

\[
a^2 - b^2 = (a - b)(a + b)
\]

Thus, \(x^2 - y^2\) can be factored as:

\[
x^2 - y^2 = (x - y)(x + y)
\]

### Step 2: Rewrite the Logarithm Expression
Substitute the factorization into the logarithm:

\[
\log_2 \left( \frac{(x - y)(x + y)}{xy} \right)
\]

### Step 3: Separate the Logarithms Using Logarithm Properties
We can use the property of logarithms that states:

\[
\log_b \left( \frac{a}{b} \right) = \log_b(a) - \log_b(b)
\]

Applying this, we get:

\[
\log_2 \left( \frac{(x - y)(x + y)}{xy} \right) = \log_2((x - y)(x + y)) - \log_2(xy)
\]

Next, apply the logarithm product rule:

\[
\log_b(ab) = \log_b(a) + \log_b(b)
\]

Using this, the expression becomes:

\[
\log_2(x - y) + \log_2(x + y) - (\log_2(x) + \log_2(y))
\]

### Step 4: Simplify the Expression
Combine like terms to simplify:

\[
\log_2(x - y) + \log_2(x + y) - \log_2(x) - \log_2(y)
\]

This is the fully expanded and simplified form of your logarithmic expression.

### Conclusion
The expression:

\[
\log_2 \left( \frac{x^2 - y^2}{xy} \right)
\]

is equivalent to:

\[
\log_2(x - y) + \log_2(x + y) - \log_2(x) - \log_2(y)
\]

Would you like more details or have any questions?

### Related Questions:
1. How do logarithmic identities work for different bases?
2. How can logarithmic expressions be simplified in different contexts?
3. What are the applications of logarithms in solving equations?
4. How does the change of base formula for logarithms work?
5. What are some common mistakes to avoid when working with logarithmic expressions?

**Tip:** When dealing with logarithmic expressions, always look for opportunities to factor or simplify the arguments to make the calculations easier.

Learn how to simplify the logarithmic expression log2((x - y)(x + y))/(xy) step-by-step with detailed explanations of logarithm properties and algebraic simplification. Suitable for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation