What is the domain of f(x) = ln(x)?

The domain of \( f(x) = \ln(x) \) is \( x > 0 \).

In more detail, the function \( f(x) = \ln(x) \) represents the natural logarithm of \( x \). The natural logarithm is only defined for positive real numbers. This means that \( x \) must be greater than zero for the function to work. If you try to take the natural logarithm of zero or a negative number, the function is undefined, and you will get an error on your calculator.

To understand why, consider the definition of the natural logarithm. The natural logarithm of a number \( x \) is the power to which the base \( e \) (approximately 2.718) must be raised to produce \( x \). Since raising \( e \) to any real number power will always result in a positive number, \( x \) must be positive. For example, \( \ln(1) = 0 \) because \( e^0 = 1 \), and \( \ln(e) = 1 \) because \( e^1 = e \).

Graphically, the function \( f(x) = \ln(x) \) has a vertical asymptote at \( x = 0 \). This means the graph approaches the y-axis but never touches or crosses it. As \( x \) gets closer to zero from the positive side, \( \ln(x) \) decreases without bound, heading towards negative infinity. Conversely, as \( x \) increases, \( \ln(x) \) increases, but at a decreasing rate.

So, when working with \( f(x) = \ln(x) \), always remember that \( x \) must be a positive number.