The given expression is:

\[
2 \log_6 4 + 3 \log_6 x - 4 \log_6 y
\]

We can simplify this by using the logarithmic properties:

1. **Power Rule**: \(\log_b (a^n) = n \log_b a\).
2. **Product Rule**: \(\log_b a + \log_b b = \log_b (a \cdot b)\).
3. **Quotient Rule**: \(\log_b a - \log_b b = \log_b \left(\frac{a}{b}\right)\).

### Step-by-step simplification:

1. Apply the **Power Rule**:
   \[
   2 \log_6 4 = \log_6 4^2 = \log_6 16
   \]
   \[
   3 \log_6 x = \log_6 x^3
   \]
   \[
   4 \log_6 y = \log_6 y^4
   \]

2. Substitute these into the original expression:
   \[
   \log_6 16 + \log_6 x^3 - \log_6 y^4
   \]

3. Apply the **Product Rule** to the first two terms:
   \[
   \log_6 (16 \cdot x^3) - \log_6 y^4
   \]

4. Finally, apply the **Quotient Rule**:
   \[
   \log_6 \left( \frac{16 \cdot x^3}{y^4} \right)
   \]

### Final simplified expression:
\[
\log_6 \left( \frac{16x^3}{y^4} \right)
\]

Would you like any additional details or further explanation?

Here are 5 related questions for practice:
1. Simplify \(3 \log_2 5 + \log_2 8 - 4 \log_2 2\).
2. Solve \(\log_3 (x^2 + 3x - 4) = 2\).
3. Expand \(\log_b (a \cdot b^2)\) using logarithmic properties.
4. Simplify \(\log_7 49 + \log_7 x - \log_7 y^2\).
5. Solve for \(x\) in \(2 \log_4 x = 3\).

**Tip**: Always keep logarithmic properties in mind to simplify complex expressions easily!

Learn how to simplify the logarithmic expression 2log6 4 + 3log6 x - 4log6 y using key logarithmic properties such as the power rule, product rule, and quotient rule. A step-by-step solution is provided for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation