The function \( F(x) = \log(x + 3) \) is a logarithmic function. Let's analyze it:

### 1. **Domain:**
   The argument of the logarithm, \( x + 3 \), must be positive because the logarithm of a non-positive number is undefined in the real number system. Therefore:
   \[
   x + 3 > 0
   \]
   \[
   x > -3
   \]
   So, the domain of \( F(x) \) is \( x > -3 \).

### 2. **Range:**
   The range of the logarithmic function is all real numbers \( (-\infty, \infty) \).

### 3. **Vertical Asymptote:**
   The function will have a vertical asymptote where the argument of the logarithm is zero:
   \[
   x + 3 = 0 \quad \Rightarrow \quad x = -3
   \]
   Therefore, the vertical asymptote is at \( x = -3 \).

### 4. **Behavior:**
   - As \( x \) approaches \( -3 \) from the right (\( x \to -3^+ \)), \( F(x) \) tends towards \( -\infty \).
   - As \( x \) increases, \( F(x) \) increases slowly and indefinitely.

### 5. **Derivative:**
   The derivative \( F'(x) \) tells us the rate of change of the function:
   \[
   F'(x) = \frac{d}{dx} \log(x + 3) = \frac{1}{x + 3}
   \]
   The derivative is positive for all \( x > -3 \), indicating that the function is increasing on its domain.

### 6. **Graph:**
   The graph of \( F(x) = \log(x + 3) \) will be a logarithmic curve that passes through \( x = -2 \) (where \( F(-2) = 0 \)), with a vertical asymptote at \( x = -3 \).

Would you like more details or have any questions?

Here are some related questions to consider:

1. How does the derivative \( F'(x) \) affect the shape of the graph?
2. What happens if the function is shifted vertically by adding a constant?
3. How would the function change if the logarithm base is different?
4. Can we express \( F(x) = \log(x + 3) \) in terms of an exponential function?
5. What is the integral of \( F(x) \) over a specific interval?
6. How does the function behave as \( x \) approaches infinity?
7. What is the inverse function of \( F(x) = \log(x + 3) \)?
8. How do transformations like stretching or reflecting affect the graph of \( F(x) \)?

**Tip:** Understanding the properties of logarithmic functions is essential for solving more complex logarithmic equations and inequalities.

Explore the logarithmic function F(x) = log(x + 3) with detailed analysis of its domain, range, asymptotes, derivative, and graph characteristics. Learn about the behavior of logarithmic functions and their applications. Suitable for high school students and anyone interested in advanced algebra and calculus concepts.

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