expand the natural logarithm of the square root of the difference of x^2 minus 36

To expand the natural logarithm of the square root of the difference between \( x^2 \) and 36, we start by expressing the problem mathematically:

\[
\ln \left( \sqrt{x^2 - 36} \right)
\]

### Step 1: Simplify the Square Root
The square root can be expressed as an exponent:

\[
\sqrt{x^2 - 36} = (x^2 - 36)^{\frac{1}{2}}
\]

### Step 2: Use the Logarithm Power Rule
The logarithm of an expression raised to a power can be simplified by bringing the exponent in front:

\[
\ln \left( (x^2 - 36)^{\frac{1}{2}} \right) = \frac{1}{2} \ln(x^2 - 36)
\]

### Final Answer
The expanded form of the natural logarithm is:

\[
\ln \left( \sqrt{x^2 - 36} \right) = \frac{1}{2} \ln(x^2 - 36)
\]

This is the expanded form of the given expression.

Would you like more details or have any questions?

Here are 5 related questions:
1. How can you further simplify \( \ln(x^2 - 36) \) if \( x^2 - 36 \) factors?
2. What are the domain restrictions of \( \ln(x^2 - 36) \)?
3. How does this expansion change if the original expression was \( \ln(\sqrt{36 - x^2}) \)?
4. How do logarithm properties apply to more complex logarithmic expressions?
5. Can you solve \( \ln(x^2 - 36) = 0 \) for \( x \)?

**Tip:** Always check the domain of the argument in logarithmic expressions to ensure the argument is positive, as the logarithm is undefined for non-positive values.

Discover how to expand the natural logarithm of the square root of x^2 - 36 step by step. This solution explores logarithmic properties, exponents, and radicals, using key formulas like the logarithm power rule. Ideal for high school students in Grades 11-12.

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