Let’s analyze and simplify the given expression, $\frac{1}{2} \log(x^2 + 3) - \log(x + 1)$:

Step-by-step Explanation:

1. Recall the logarithmic properties:

   • $a \log b = \log(b^a)$
   • $\log b - \log c = \log \left(\frac{b}{c}\right)$

2. Simplify $\frac{1}{2} \log(x^2 + 3)$: Using the power property, $\frac{1}{2} \log(x^2 + 3) = \log\left((x^2 + 3)^{1/2}\right) = \log\left(\sqrt{x^2 + 3}\right)$.

3. Combine the terms: The expression becomes: $\log\left(\sqrt{x^2 + 3}\right) - \log(x + 1)$ Using the subtraction rule for logarithms: $\log\left(\frac{\sqrt{x^2 + 3}}{x + 1}\right)$

Final Expression:

$\log\left(\frac{\sqrt{x^2 + 3}}{x + 1}\right)$

Let me know if you have more questions or need further clarification!

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5 Related Questions:

1. How can logarithmic properties be used to simplify more complex expressions?
2. What happens if the base of the logarithm changes in the above problem?
3. How do you solve equations involving logarithms like this one?
4. Can you explain how the natural logarithm (ln) differs from log in base 10?
5. How do logarithmic functions relate to exponential functions?

Tip:

Always verify the domain of the expression, especially for logarithmic terms, to ensure no undefined operations occur (e.g., logarithm of a negative number or zero).