To graph a logarithmic function, let's start by considering a general logarithmic function of the form:

\[ y = \log_b(x) \]

Here:
- \( b \) is the base of the logarithm.
- \( x \) is the input variable.

### Steps to Graph a Logarithmic Function:

1. **Identify the Base**: Determine the base \( b \). If \( b > 1 \), the function is increasing; if \( 0 < b < 1 \), the function is decreasing.

2. **Determine the Domain**: The function is only defined for \( x > 0 \), meaning that the graph exists only in the first quadrant.

3. **Find Key Points**:
   - **When \( x = 1 \)**, \( y = \log_b(1) = 0 \) (since any logarithm of 1 is 0).
   - **When \( x = b \)**, \( y = \log_b(b) = 1 \).
   - **When \( x = \frac{1}{b} \)**, \( y = \log_b\left(\frac{1}{b}\right) = -1 \).

4. **Asymptote**: The graph has a vertical asymptote at \( x = 0 \) because as \( x \) approaches 0 from the right, the logarithmic function approaches negative infinity.

5. **Sketch the Graph**:
   - Plot the key points found in step 3.
   - Draw the vertical asymptote at \( x = 0 \).
   - Sketch the curve passing through the key points, making sure it approaches the asymptote but never crosses it.

### Example: Graph \( y = \log_2(x) \)

1. **Base**: \( b = 2 \) (so the function is increasing).
2. **Key Points**:
   - \( (1, 0) \)
   - \( (2, 1) \)
   - \( \left(\frac{1}{2}, -1\right) \)

3. **Vertical Asymptote**: At \( x = 0 \).

4. **Sketch**: Plot the points and draw the curve approaching the asymptote at \( x = 0 \).

Would you like me to plot this graph for you, or do you have any questions?

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### Related Questions:
1. How does changing the base \( b \) affect the graph of the logarithmic function?
2. What is the relationship between logarithmic and exponential functions?
3. How do you find the inverse of a logarithmic function?
4. Can logarithmic functions have negative outputs?
5. How do you solve equations involving logarithms?

**Tip**: Remember that the logarithm function is the inverse of the exponential function. Understanding this relationship can help in solving and graphing these functions effectively.

Learn how to graph a logarithmic function with this comprehensive guide. Understand key concepts like logarithm properties, vertical asymptotes, and how to plot crucial points for accurate graphing. Suitable for students in Grades 10-12.

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