To graph a logarithmic function, we follow a series of steps. Here’s a breakdown of the process for graphing a basic logarithmic function, like $y = \log_b(x)$, where $b$ is the base of the logarithm (typically $b = 10$ for common logarithms, or $b = e$ for natural logarithms).

Steps to Graph Logarithmic Functions:

1. Understand the Basic Shape:

• The graph of $y = \log_b(x)$ is a curve that passes through the point $(1, 0)$, since $\log_b(1) = 0$ for any base $b > 0$.
• The graph is undefined for $x \leq 0$. So, it exists only in the right half of the coordinate plane (positive x-axis).
• As $x$ approaches 0 from the right, $y$ will approach $-\infty$.
• As $x$ increases, the graph gradually increases, but at a decreasing rate. It is a slowly increasing curve.

2. Identify Key Points:

For $y = \log_b(x)$, some easy points to calculate are:

• $\log_b(1) = 0$, so the graph passes through $(1, 0)$.
• $\log_b(b) = 1$, so the graph passes through $(b, 1)$.
• If you have other values for $x$, you can substitute them into the logarithmic equation to find corresponding $y$-values.

3. Plot Points:

• Choose several values for $x$ (for example, 0.1, 1, 10, 100) and calculate their logarithms.
• Plot these points on the coordinate plane.

4. Draw the Curve:

• Once enough points are plotted, sketch the curve. Make sure it approaches $-\infty$ as $x$ approaches 0 and gradually increases as $x$ becomes large.

Example:

Let’s graph $y = \log_2(x)$:

1. Key Points:

   • $\log_2(1) = 0$, so the graph passes through $(1, 0)$.
   • $\log_2(2) = 1$, so the graph passes through $(2, 1)$.
   • $\log_2(4) = 2$, so the graph passes through $(4, 2)$.
   • As $x$ approaches 0, $y$ becomes negative and heads toward $-\infty$.

2. Plot these points on a graph, and you’ll see that the curve starts from near $-\infty$ at $x = 0$, passes through $(1, 0)$, and increases slowly.

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Important Tips:

• Vertical Asymptote: The graph has a vertical asymptote at $x = 0$ (the y-axis), meaning the graph never touches or crosses the y-axis.
• Transformations: If the equation is in the form $y = \log_b(x - h) + k$, this means there is a horizontal shift by $h$ units and a vertical shift by $k$ units. The base $b$ will also affect the steepness of the curve.

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Would you like to explore transformations, specific graph examples, or work through some related problems? Let me know!

Here are 5 questions you can consider:

1. How does changing the base $b$ affect the graph of the logarithmic function?
2. What happens when the function is in the form $y = \log_b(x + h)$?
3. Can you find the asymptote of a logarithmic function and explain its significance?
4. How do you graph natural logarithms (base $e$) and compare them to common logarithms (base 10)?
5. How can logarithmic graphs be applied in real-world problems, such as population growth or sound intensity?

Tip: For graphing logarithms, it’s always useful to make a table of values for $x$ and calculate corresponding $y$-values to plot accurate points.