91Ó°ÊÓ

A logarithmic function is the inverse of an exponential function. In general, if you have a function of the form \( y = \ln(x) \), it means the power to which the base of the natural logarithm, e, must be raised to produce that number x. Here, e is a constant approximately equal to 2.71828. The natural logarithm (ln) specifically has the base e and is commonly used in various mathematical, physical, and engineering problems.

Logarithmic functions have unique properties:

• They pass through the point (1,0) since \(\ln(1) = 0\).
• The function is undefined for non-positive values (i.e., x ≤ 0).
• The logarithmic curve always passes from the negative y-axis to the positive y-axis, increasing slowly.

To graph these functions, you primarily need to understand shifts, stretches, and reflections. In this exercise, we focus on the natural logarithm function, which is slightly shifted due to its argument being (x-1). This shift moves the standard \(\ln(x)\) function to the right by 1 unit.

The domain and range are critical to understanding any function.

**Domain**
The domain of a function consists of all possible input values (x-values) that allow the function to work. For a natural logarithm function like \( g(x) = \ln (x-1) \), the argument x-1 must be greater than zero. This is because you cannot take the logarithm of zero or a negative number.

Therefore, we set up the inequality:
\( x - 1 > 0 \) or \( x > 1 \). In interval notation, the domain is written as \( (1, \infty) \). In simpler terms, x must be greater than 1.

**Range**
The range of a function includes all possible output values (y-values). For the natural logarithm function, the range is all real numbers. There is no restriction on the y-values because logarithms can extend infinitely in both the positive and negative directions.

Therefore, the range of \( g(x) = \ln (x-1) \) is \( (-\infty, \infty) \). So, even if we shift the graph, the range of the natural logarithm remains the same.

Graphing logarithmic functions can be made simple with some basic guidelines:

• First, identify the base function. Here it is \( \ln(x) \).
• Determine any transformations such as shifts, stretches, or compressions. In our case, \( \ln (x-1) \) involves a horizontal shift to the right by 1 unit.
• Locate the vertical asymptote, which is a value x approaches but never actually reaches. For \( \ln (x-1) \), the asymptote is at \( x = 1 \).

Next, plot some key points to guide your graph. For example:

• When x = 2, \( g(2) = \ln (2-1) = 0 \).
• When x = 3, \( g(3) = \ln (3-1) = \ln 2 \).

Finally, sketch the curve, ensuring it gets closer and closer to the asymptote at x = 1, but never touching it, and grows without bound as x increases. Remember, the curve should continue indefinitely towards positive and negative infinity in the y-direction. This visual representation is crucial for understanding how logarithmic functions behave and interact with their domain restrictions.