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Understanding the characteristics of logarithmic functions is essential for graphing them accurately. A logarithm function, represented as \(y = \text{log}_b(x)\), ties together three key components: the output \(y\), the input \(x\), and the base \(b\). Logarithmic properties tell us that the log of a product is the sum of the logs \(\text{log}_b(xy) = \text{log}_b(x) + \text{log}_b(y)\), the log of a quotient is the difference \(\text{log}_b(\frac{x}{y}) = \text{log}_b(x) - \text{log}_b(y)\), and the log of a power is the exponent times the log \(\text{log}_b(x^k) = k \cdot \text{log}_b(x)\).

Additionally, the function will output 0 when \(x = b\), as \(\text{log}_b(b) = 1\). The logarithm is undefined for \(x\) values that are less than or equal to 0. It's crucial to remember these rules when interpreting and graphing logarithmic functions.

When dealing with logarithms where the base is less than 1, it's important to note that the graph will exhibit a specific behavior. For a function like \(y = \text{log}_{1/3}(x)\), where the base \(\frac{1}{3}\) is between 0 and 1, the shape of the graph will be a decreasing function as \(x\) increases. This decrease is due to the simple fact that raising a fraction to a higher power results in a smaller number, and the logarithm of a smaller number is less.

This concept is clearly visible when plotting points: as we move to the right along the \(x\)-axis to higher values of \(x\), the corresponding \(y\) values become more negative, hence the downward slope of the graph. Grasping this pattern is vital for effectively sketching logarithmic graphs with bases less than 1.

Sketching logarithmic graphs begins with understanding the function's behavior and plotting key points. For the function \(y = \text{log}_{1/3}(x)\), you first find some key points to act as a guideline. These are calculated by choosing various \(x\)-values and determining their corresponding \(y\)-values. As shown in the exercise, points such as \((1, 0)\), \((3, -1)\), and \((9, -2)\) are derived from picking values for \(x\) and using the logarithm to calculate \(y\).

Once you plot these points, you connect them with a smooth curve, ensuring to reflect the characteristic shapes indicated by the base of the logarithm. If it's less than 1, as in our case, the graph will decrease and have a shape that bends downward as it moves to the right.

One of the defining characteristics of the logarithmic graph is the vertical asymptote. This is a vertical line that the graph approaches but never actually touches or crosses. For the general form \(y = \text{log}_b(x)\), the vertical asymptote is always at \(x = 0\). It represents the boundary beyond which the function is not defined, since logarithms are undefined for 0 and negative numbers.

When you sketch the graph of a logarithmic function with \(b < 1\), such as \(y = \text{log}_{1/3}(x)\), the vertical asymptote at \(x = 0\) stays consistent. This is important to note when drawing your graph, as the function's curve should approach this line on the left side but not cross it, giving a clear visual indicator of the function’s domain.