To match the functions with their graphs, let's break down the key characteristics of each logarithmic function:

1. $f(x) = -\ln(x)$

• This is a reflection of $\ln(x)$ across the x-axis.
• The graph approaches $-\infty$ as $x \to 0^+$ and approaches 0 from below as $x \to \infty$.
• The domain is $x > 0$.

2. $f(x) = 2 + \ln(x)$

• This is a vertical shift of $\ln(x)$ upwards by 2 units.
• The graph behaves like $\ln(x)$, but every point is 2 units higher.
• The domain is $x > 0$, and it has a vertical asymptote at $x = 0$.

3. $f(x) = \ln(2 - x)$

• This is a reflection of $\ln(x)$, horizontally shifted right by 2 units.
• The graph is defined for $x < 2$, and has a vertical asymptote at $x = 2$.
• As $x \to 2^-$, the graph approaches $-\infty$, and as $x \to -\infty$, the graph rises slowly.

4. $f(x) = \ln(x - 2)$

• This is a horizontal shift of $\ln(x)$ to the right by 2 units.
• The graph is defined for $x > 2$, with a vertical asymptote at $x = 2$.
• As $x \to 2^+$, the graph approaches $-\infty$, and as $x \to \infty$, the graph rises slowly.

5. $f(x) = -\ln(-x)$

• This is a reflection of $\ln(-x)$ across the x-axis.
• The graph is defined for $x < 0$, with a vertical asymptote at $x = 0$.
• As $x \to 0^-$, the graph approaches $+\infty$, and as $x \to -\infty$, the graph approaches 0 from above.

Now, you can match these descriptions with the corresponding graphs by analyzing the domain, asymptotes, and general behavior of each function.

Do you have the graphs available for me to match them precisely, or should I help further with how to approach graph identification?

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Follow-up questions:

1. What is the domain of $\ln(x)$, and how does it impact transformations?
2. How does reflecting a logarithmic function across the x-axis affect its shape?
3. How do vertical and horizontal shifts affect the graph of logarithmic functions?
4. What is the significance of vertical asymptotes in the behavior of logarithmic functions?
5. How can you determine if a function's graph is defined for positive or negative values of $x$?

Tip:

Logarithmic functions are undefined for non-positive values of their arguments, so always check the domain carefully when shifting or reflecting graphs.