The graph of the function $y = \ln(x)$ is characterized by a gently increasing curve that passes through the point $(1, 0)$.

The natural logarithm function, $\ln(x)$, is defined exclusively for positive values of $x$. Consequently, its graph exists solely in the right half of the coordinate plane, where $x > 0$. As $x$ approaches $0$ from the right, the value of $y = \ln(x)$ decreases indefinitely, tending towards negative infinity. This behavior results in a vertical asymptote at $x = 0$, indicating that the graph approaches the y-axis but never intersects it.

At the point $x = 1$, the natural logarithm of $1$ is $0$, which means the graph intersects at $(1, 0)$. This point is crucial for accurately sketching the graph. As $x$ increases beyond $1$, the value of $y = \ln(x)$ continues to rise, albeit at a decreasing rate. This implies that the graph ascends more slowly as $x$ grows larger.

The overall shape of the graph is a smooth, continuous curve that starts from negative infinity as $x$ approaches $0$, passes through the point $(1, 0)$, and continues to rise gradually as $x$ increases. Notably, the graph never touches or crosses the y-axis and extends infinitely to the right. Familiarity with these characteristics enables one to effectively sketch the graph of $y = \ln(x)$ and analyze its behavior in various mathematical contexts.