\ln(x) + \ln(2x) = \ln(x \cdot 2x) = \ln(2x^2)

Yes, the logarithmic equation you provided is correct and demonstrates the properties of logarithms.

Here's the step-by-step breakdown:

### Given expression:
\[
\ln(x) + \ln(2x)
\]

#### Step 1: Use the logarithmic property \(\ln(a) + \ln(b) = \ln(a \cdot b)\)
\[
\ln(x) + \ln(2x) = \ln(x \cdot 2x)
\]

#### Step 2: Simplify the product inside the logarithm:
\[
\ln(x \cdot 2x) = \ln(2x^2)
\]

So, the equation:
\[
\ln(x) + \ln(2x) = \ln(2x^2)
\]
is valid by using logarithmic properties.

Would you like more detailed explanations on logarithmic properties, or do you have any further questions?

### 5 Related Questions:
1. How would you solve \(\ln(2x^2) = 4\)?
2. What are the general properties of logarithms?
3. How can we differentiate \(\ln(2x^2)\) with respect to \(x\)?
4. How do we solve logarithmic equations of the form \(\ln(x) + \ln(x + 1) = 5\)?
5. What happens if \(x\) is negative in \(\ln(x)\)?

### Tip:
When working with logarithmic equations, always remember that \(\ln(a \cdot b) = \ln(a) + \ln(b)\), which is one of the most commonly used properties for simplifying expressions.

Understand how to simplify the logarithmic equation \ln(x) + \ln(2x) = \ln(2x^2) by using logarithmic properties. This step-by-step guide explains key logarithmic concepts, perfect for students in Grades 9-12.

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