Plot the following.
`y=|log_(2)|x||`

Answer

To plot the function \( y = |\log_2 |x|| \), we will follow these steps:

### Step 1: Understand the function components
The function consists of two parts: the logarithm base 2 of the absolute value of \( x \), and the absolute value of that logarithm.

### Step 2: Analyze \( y = \log_2 |x| \)
- The logarithm function \( \log_2 |x| \) is defined for \( |x| > 0 \) (i.e., \( x \neq 0 \)).
- The function \( \log_2 |x| \) will be negative for \( 0 < |x| < 1 \) and positive for \( |x| > 1 \).
- The key points to note are:
- \( \log_2 |1| = 0 \)
- \( \log_2 |2| = 1 \)
- As \( |x| \) approaches 0, \( \log_2 |x| \) approaches negative infinity.

### Step 3: Plot \( y = \log_2 |x| \)
- For \( x > 0 \):
- The graph will start from \( (1, 0) \) and rise to the right, passing through \( (2, 1) \).
- For \( x < 0 \):
- The graph will be symmetric about the y-axis because of the absolute value, so it will also start from \( (1, 0) \) and rise to the left, passing through \( (-2, 1) \).

### Step 4: Analyze \( y = |\log_2 |x|| \)
- The absolute value of the logarithm means that any negative values will be reflected above the x-axis.
- For \( 0 < |x| < 1 \), \( \log_2 |x| \) is negative, so \( |\log_2 |x|| \) will be positive.
- For \( |x| > 1 \), \( \log_2 |x| \) is positive, so \( |\log_2 |x|| \) remains the same.

### Step 5: Plot the final graph
- For \( 0 < x < 1 \):
- The graph will go from \( (0, \infty) \) to \( (1, 0) \).
- For \( x = 1 \):
- The graph touches the point \( (1, 0) \).
- For \( 1 < x < 2 \):
- The graph will rise from \( (1, 0) \) to \( (2, 1) \).
- For \( x = -1 \):
- The graph will touch the point \( (-1, 0) \).
- For \( -2 < x < -1 \):
- The graph will rise from \( (-2, 1) \) to \( (-1, 0) \).
- The overall graph will be symmetric about the y-axis.

### Final Graph
The final graph of \( y = |\log_2 |x|| \) will have the following characteristics:
- It will touch the x-axis at \( x = 1 \) and \( x = -1 \).
- It will rise to infinity as \( x \) approaches 0 from either side.
- It will rise to \( (2, 1) \) and \( (-2, 1) \).