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.