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Let's work through the problem given in the image.

### Function: 
\[
h(x) = -\log(3x - 5) + 7
\]

### 1. **Domain:**
The domain of a logarithmic function is determined by the argument of the logarithm being positive. Therefore, for this function:

\[
3x - 5 > 0 \implies x > \frac{5}{3}
\]

Thus, the domain in interval notation is:
\[
\boxed{\left( \frac{5}{3}, \infty \right)}
\]

### 2. **Vertical Asymptote:**
The vertical asymptote occurs where the argument of the logarithm is equal to zero, which happens at:
\[
3x - 5 = 0 \implies x = \frac{5}{3}
\]

So, the vertical asymptote is at:
\[
\boxed{x = \frac{5}{3}}
\]

### 3. **End Behavior:**
Now, let's analyze the behavior of the function:

- **As \( x \to \frac{5}{3}^+ \) (approaching the vertical asymptote from the right):**
  When \( x \) is close to \( \frac{5}{3} \), \( 3x - 5 \) approaches zero from the positive side, making \( \log(3x - 5) \to -\infty \). Since there is a negative sign in front of the logarithm, we get:
  \[
  h(x) = -\log(3x - 5) + 7 \to \infty
  \]
  Hence, as \( x \to \frac{5}{3}^+ \):
  \[
  h(x) \to \infty
  \]

- **As \( x \to \infty \):**
  As \( x \to \infty \), \( 3x - 5 \) grows without bound, and \( \log(3x - 5) \to \infty \). With the negative sign, we have:
  \[
  h(x) = -\log(3x - 5) + 7 \to 7
  \]
  Hence, as \( x \to \infty \):
  \[
  h(x) \to 7
  \]

### Final Answer:

- **Domain**: \( \boxed{ \left( \frac{5}{3}, \infty \right) } \)
- **Vertical Asymptote**: \( \boxed{ x = \frac{5}{3} } \)
- **Asymptotic Behavior**:
  - As \( x \to \frac{5}{3}^+ \), \( h(x) \to \infty \)
  - As \( x \to \infty \), \( h(x) \to 7 \)

Would you like more details on any part of this? Here are some related questions to explore further:

1. How do logarithmic transformations affect the behavior of functions near vertical asymptotes?
2. How does the negative sign in front of a logarithmic function impact the end behavior?
3. What are some general methods to determine the range of a logarithmic function?
4. How do you identify horizontal asymptotes in functions with logarithmic terms?
5. How can you graph \( h(x) = -\log(3x - 5) + 7 \) using the information found?

**Tip:** When determining the domain of a logarithmic function, always check for when the argument inside the logarithm is positive!

State the domain, vertical asymptote, and end behavior of the function h(x) = -log(3x - 5) + 7.

Learn how to determine the domain, vertical asymptote, and end behavior of the logarithmic function h(x) = -log(3x - 5) + 7. This problem covers key logarithmic function concepts and their asymptotic behavior. Suitable for students in Grades 9-12.

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