To determine the correct transformation for the function $h(x) = -\ln(x)$ from the base function $f(x) = \ln(x)$:

Step-by-step Explanation:

1. Understanding the Transformation:

   • The negative sign in front of $\ln(x)$ indicates that the graph of $f(x) = \ln(x)$ is reflected about the x-axis.
   • This means that all $y$-values of the original graph will now become their opposites, flipping the graph vertically.

2. Equation of the Asymptote:

   • The vertical asymptote remains the same as for $f(x) = \ln(x)$, which is $x = 0$.
   • This is because the logarithmic function is undefined for $x \leq 0$.

3. Domain and Range:

   • Domain: Since $x > 0$ for $\ln(x)$, the domain of $h(x) = -\ln(x)$ is still $x > 0$.
   • Range: After reflection, the range changes to $(-\infty, 0)$, as the graph is flipped below the x-axis.

4. Correct Choice:

   • The correct answer is B: The graph of $f(x) = \ln(x)$ should be reflected about the x-axis.

Let me know if you want a deeper breakdown or visualizations!

Here are 5 related questions to explore this topic further:

1. What transformations occur when you shift $\ln(x)$ left or right?
2. How does the range of $\ln(x)$ change with a vertical stretch or compression?
3. What is the impact of reflecting $\ln(x)$ about the y-axis instead?
4. How would you write the function if $\ln(x)$ were shifted 2 units downward?
5. What is the domain of $\ln(-x)$, and how does its graph compare to $\ln(x)$?

Tip: Always check how negative signs and coefficients affect both the domain and range of logarithmic functions!