https://file.aiquickdraw.com/mathbot/file/1729309511403v5h5f8ph.jpg

To combine the expression \( 4 \ln(x) + 2 \ln(y) \) into a single logarithm, we can use the properties of logarithms. Here's how to do it step by step:

1. **Use the Power Rule**: The logarithmic power rule states that \( a \ln(b) = \ln(b^a) \). Apply this rule to both terms:
   \[
   4 \ln(x) = \ln(x^4) \quad \text{and} \quad 2 \ln(y) = \ln(y^2)
   \]

2. **Use the Product Rule**: The logarithmic product rule states that \( \ln(a) + \ln(b) = \ln(a \cdot b) \). Now apply this rule to combine the two logarithms:
   \[
   \ln(x^4) + \ln(y^2) = \ln(x^4 \cdot y^2)
   \]

Thus, the expression simplifies to:
\[
\ln(x^4 \cdot y^2)
\]

This is the expression written as a single logarithm.

Would you like more details or have any questions about these steps?

Here are 5 related questions to explore this topic further:
1. How do you apply the logarithmic quotient rule?
2. Can logarithms of negative numbers be calculated in the real number system?
3. How do you differentiate a logarithmic function?
4. What are the key differences between natural logs and common logs?
5. How can logarithms be used to solve exponential equations?

**Tip**: Always remember that when working with logarithmic properties, the base of the logarithm matters, but these rules (power, product, and quotient) apply for any logarithm, including natural logarithms.

4 ln(x) + 2 ln(y): Use the properties of logarithms to write the expression as a single logarithm.

Learn how to simplify the expression 4 ln(x) + 2 ln(y) into a single logarithm by applying the power and product rules of logarithms. This step-by-step solution explains key concepts and formulas for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation