Draw the graph of y = |log x|.

To graph the function \( y = | \log x | \), we first need to understand the components of the equation.

Understanding the Logarithm

The logarithm function \( \log x \) is defined for \( x > 0 \). It represents the power to which the base (usually 10 or e) must be raised to obtain \( x \).

Behavior of \( \log x \)

• As \( x \) approaches 0 from the right, \( \log x \) approaches negative infinity.
• At \( x = 1 \), \( \log 1 = 0 \).
• As \( x \) increases, \( \log x \) increases without bound.

Applying the Absolute Value

The absolute value function \( | \log x | \) transforms all negative values of \( \log x \) into positive values. This means:

• For \( 0 < x < 1 \), \( | \log x | = -\log x \).
• For \( x = 1 \), \( | \log x | = 0 \).
• For \( x > 1 \), \( | \log x | = \log x \).

Sketching the Graph

Now, let's visualize the graph:

• For \( 0 < x < 1 \), the graph will be a curve that starts from positive infinity and approaches 0 as \( x \) approaches 1.
• At \( x = 1 \), the graph touches the x-axis.
• For \( x > 1 \), the graph will rise steadily as \( x \) increases.

Key Points to Plot

• Point (0.1, 1): \( | \log 0.1 | = 1 \)
• Point (1, 0): \( | \log 1 | = 0 \)
• Point (10, 1): \( | \log 10 | = 1 \)

In summary, the graph of \( y = | \log x | \) consists of two parts: a decreasing curve from positive infinity to the origin at \( x = 1 \), and an increasing curve for \( x > 1 \). This creates a "V" shape with its vertex at the point (1, 0).