Draw the graph of y = log |x| when the graph of y = log (x) is known.

To understand the graph of \( y = \log |x| \), we first need to consider the graph of \( y = \log(x) \). The logarithmic function \( y = \log(x) \) is defined only for positive values of \( x \) and has the following characteristics:

Key Features of \( y = \log(x) \)

• Domain: \( x > 0 \)
• Range: All real numbers
• Intercept: (1, 0), since \( \log(1) = 0 \)
• Asymptote: Vertical line at \( x = 0 \)

Now, when we consider \( y = \log |x| \), we need to account for both positive and negative values of \( x \). The absolute value allows the logarithm to be defined for negative \( x \) as well.

Graphing \( y = \log |x| \)

The graph of \( y = \log |x| \) can be derived as follows:

• For \( x > 0 \), the graph is identical to \( y = \log(x) \).
• For \( x < 0 \), the graph mirrors the positive side across the y-axis. This means that \( y = \log |x| \) will have the same values as \( y = \log(x) \) but will be reflected in the second quadrant.

Visual Representation

In summary, the graph of \( y = \log |x| \) consists of two parts:

• The right side (for \( x > 0 \)) follows the curve of \( y = \log(x) \).
• The left side (for \( x < 0 \)) mirrors this curve, creating a symmetric appearance about the y-axis.

This results in a continuous graph that extends infinitely in both directions along the y-axis, with vertical asymptotes at \( x = 0 \).