Describe the general shape and key features of the graph of a logarithmic function, including any asymptotes.

The graph of a logarithmic function, such as \( y = \log_b(x) \), typically has a characteristic "swooping" shape that rises slowly from the left and continues to expand to the right. The key feature is that it never touches or crosses the vertical line at \( x = 0 \), which acts as a vertical asymptote. As \( x \) approaches zero from the right, the function heads toward negative infinity. The graph also steadily increases without bound as \( x \) increases, but the growth rate diminishes over time, giving it a logarithmic curve. In practical terms, logarithmic functions are often used in areas such as sound intensity (decibels), pH levels in chemistry, and the Richter scale for earthquakes. For example, a small increase in the Richter scale represents a significant increase in actual energy release during an earthquake. This makes logarithmic scales invaluable in comparing quantities that span several orders of magnitude, enabling us to grasp the significance of very large or very small values in a more relatable manner.