The given expression is:

$y = \ln \left( \frac{\sqrt{x^2 + 4 - x^2}}{\sqrt{x^2 + 4 + x^2}} \right)$

Step-by-step simplification:

1. Simplify the square roots:

   • The numerator is $\sqrt{x^2 + 4 - x^2}$: Simplify $x^2 - x^2 = 0$, so the numerator becomes $\sqrt{4} = 2$.
   • The denominator is $\sqrt{x^2 + 4 + x^2}$: Simplify $x^2 + x^2 = 2x^2$, so the denominator becomes $\sqrt{2x^2 + 4}$.

   Thus, the expression becomes: $y = \ln \left( \frac{2}{\sqrt{2x^2 + 4}} \right)$

2. Factorize the denominator: In $\sqrt{2x^2 + 4}$, factor out 2 from the square root: $\sqrt{2x^2 + 4} = \sqrt{2(x^2 + 2)} = \sqrt{2} \cdot \sqrt{x^2 + 2}$ Substituting this back, the expression becomes: $y = \ln \left( \frac{2}{\sqrt{2} \cdot \sqrt{x^2 + 2}} \right)$

3. Simplify further: The fraction simplifies as: $\frac{2}{\sqrt{2} \cdot \sqrt{x^2 + 2}} = \frac{2}{\sqrt{2}} \cdot \frac{1}{\sqrt{x^2 + 2}}$ Simplify $\frac{2}{\sqrt{2}} = \sqrt{2}$, so: $y = \ln \left( \frac{\sqrt{2}}{\sqrt{x^2 + 2}} \right)$

4. Separate the logarithms: Using the logarithmic property $\ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b)$, split the expression: $y = \ln(\sqrt{2}) - \ln(\sqrt{x^2 + 2})$ Simplify further: $\ln(\sqrt{2}) = \frac{1}{2} \ln(2), \quad \ln(\sqrt{x^2 + 2}) = \frac{1}{2} \ln(x^2 + 2)$ Thus: $y = \frac{1}{2} \ln(2) - \frac{1}{2} \ln(x^2 + 2)$

Final Answer:

$y = \frac{1}{2} \ln(2) - \frac{1}{2} \ln(x^2 + 2)$

Would you like a deeper explanation of any step?

Related Questions:

1. How do you simplify logarithmic expressions involving fractions?
2. What are common logarithmic properties used in simplification?
3. How can $\ln(a) - \ln(b)$ be combined into one expression?
4. Why does $\ln(\sqrt{a})$ equal $\frac{1}{2} \ln(a)$?
5. What are the applications of logarithms in real-world problems?

Tip:

When simplifying square roots, always check if factoring can make the expression cleaner before substituting values back into the equation.