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When sketching the graph of a logarithmic function like \(y = \log_{3} x\), it's all about understanding its basic shape and properties. First, note the curve is only defined for values of \(x > 0\); this is because you cannot take the logarithm of a non-positive number. The function is continuous and increases slowly as \(x\) grows larger. For a sketch:

• Start by drawing a vertical asymptote line at \(x = 0\)
• Make note of important points such as the x-intercept when \(x = 1\)
• Include other points like \( (3,1) \) which help with representation
• Finally, show the increasing nature of the curve as x becomes larger

Use these steps to create a smooth curve that's close to the x-axis on the left and rises gradually on the right.

To find where the graph of \(y = \log_{3} x\) crosses the x-axis, we need to determine the x-intercept. This is the point where the value of \(y\) is zero. This can be simplified using the equation:\[0 = \log_{3} x\]Converting this logarithmic equation into its exponential form gives:\[3^0 = x\]Thus, \(x = 1\). This tells us that when \(y\) equals zero, \(x\) is 1, indicating the graph crosses the x-axis at this point. The x-intercept is a foundational point for sketching the function and showing its behavior at \(x = 1\).

The vertical asymptote is a key feature of any logarithmic graph, including \(y = \log_{3} x\). It acts as a boundary that the graph approaches but never crosses. For the function \(y = \log_{3} x\), this vertical asymptote is located at \(x = 0\). Why is there an asymptote at \(x = 0\)?

• Because logarithmic functions are undefined for non-positive numbers.
• As \(x\) approaches 0 from the positive side, the y-values sharply decrease.

Remember, when sketching this function, the graph nears the line \(x = 0\) but never touches or crosses it.

Understanding the graph characteristics of \(y = \log_{3} x\) can enhance your sketch and comprehension of logarithmic behaviors. Here are some core aspects to consider:- **Domain**: The function is only defined for \(x > 0\). There are no values of \(y\) when \(x\) is zero or negative.- **Range**: Since the graph can extend indefinitely upward or downward based on \(x\), the range is all real numbers.- **Curve Behavior**: - It rises gently but steadily as \(x\) increases beyond 1. - It never touches or crosses the vertical asymptote.- **Sympathetic Points**: - The graph is symmetric in that it is continuously increasing.These characteristics together allow the depiction of a typical logarithmic curve, maintaining the essential properties within its unbounded domain and range.