The graph in the image displays three logarithmic functions, and you need to match the functions with their respective graph colors based on their behavior.

Let's analyze the logarithmic functions:

1. $f(x) = \log_{10}(x)$
2. $f(x) = \log_2(x)$
3. $f(x) = \ln(x)$ (natural logarithm)

Behavior of Logarithmic Functions:

• Base 10 ($\log_{10}(x)$): This is slower-growing compared to smaller bases like 2 or the natural log.
• Base 2 ($\log_2(x)$): This grows slightly faster than the log base 10 but slower than the natural log.
• Natural log ($\ln(x)$): This function grows faster than the previous two.

Now let's match them to the graph:

1. Green (solid line): This line grows the fastest, so it represents the natural logarithm $f(x) = \ln(x)$.
2. Red (long dash): The red line grows slower than the green line but faster than the blue one, indicating it corresponds to the base 2 logarithm $f(x) = \log_2(x)$.
3. Blue (short dash): The blue line grows the slowest, matching the base 10 logarithm $f(x) = \log_{10}(x)$.

Final Answers:

• Red (long dash): $f(x) = \log_2(x)$
• Blue (short dash): $f(x) = \log_{10}(x)$
• Green (solid): $f(x) = \ln(x)$

Would you like further explanation on logarithmic growth or how the functions behave? Let me know!

Here are some follow-up questions related to logarithmic functions:

1. What is the inverse of a logarithmic function?
2. How do logarithms relate to exponential functions?
3. How does the change of base formula work for logarithms?
4. What are some practical applications of logarithmic functions?
5. How do vertical asymptotes occur in logarithmic functions?

Tip: When graphing logarithmic functions, remember that they all have a vertical asymptote at $x = 0$ and increase without bound as $x$ increases.