Graph the parent function x
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g(x)=log
0.5
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(x)

The function you're asking to graph is \( g(x) = \log_{0.5}(x) \). This is a logarithmic function with a base of 0.5. The parent function for logarithms is generally \( \log_b(x) \), where \( b \) is the base.

### Key Characteristics of \( g(x) = \log_{0.5}(x) \):

1. **Domain**: The function is only defined for \( x > 0 \).
2. **Range**: The range is all real numbers \( (-\infty, \infty) \).
3. **Intercept**: It crosses the x-axis at \( x = 1 \) because \( \log_{0.5}(1) = 0 \).
4. **Asymptote**: There is a vertical asymptote at \( x = 0 \).
5. **Behavior**: Since the base \( 0.5 \) is between 0 and 1, the function is decreasing. This means as \( x \) increases, \( g(x) \) decreases.

I will now graph this function.It seems there was an issue generating the graph. I will try again.It seems the graph is still not generating correctly. I will describe the graph instead:

### Description of the Graph:
- **Shape**: The graph of \( g(x) = \log_{0.5}(x) \) starts high on the left (as \( x \) approaches 0 from the positive side) and decreases as \( x \) increases, curving downward as it approaches the x-axis.
- **Asymptote**: The vertical asymptote at \( x = 0 \) means the graph goes infinitely down as it gets closer to the y-axis from the right.
- **Intercept**: The graph crosses the x-axis at \( x = 1 \), since \( \log_{0.5}(1) = 0 \).
- **Decreasing Nature**: The graph decreases as \( x \) increases because the base of the logarithm is between 0 and 1.

Would you like to explore this further, or perhaps try another question?

### Related Questions:
1. What is the general shape of a logarithmic function when the base is less than 1?
2. How does changing the base of a logarithm affect the graph?
3. What happens to the graph of \( g(x) = \log_{b}(x) \) if \( b > 1 \)?
4. How would the graph change if the function were \( g(x) = -\log_{0.5}(x) \)?
5. Can you explain why there is a vertical asymptote at \( x = 0 \)?
6. How does the logarithm base relate to exponential functions?
7. What are some real-world applications of logarithmic functions?
8. How would you solve an equation involving \( \log_{0.5}(x) \)?

### Tip:
When graphing logarithmic functions, always consider the base. Bases between 0 and 1 will yield decreasing functions, while bases greater than 1 will yield increasing functions.

Explore how to graph the logarithmic function g(x) = log base 0.5 of x. Learn about its key characteristics, including domain, range, intercepts, and behavior. This detailed explanation covers why there's a vertical asymptote at x = 0 and how changing the base affects the graph. Suitable for high school students studying logarithmic functions.

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