The number of solutions of the equation `e ^(x) - log |x|=0` is :

Options

A. 0
B. 1
C. 2
D. 5

Answer

To find the number of solutions of the equation \( e^x - \log |x| = 0 \), we can rewrite it as:

\[ e^x = \log |x| \]

### Step 1: Analyze the functions
We will analyze the two functions \( f(x) = e^x \) and \( g(x) = \log |x| \).

### Step 2: Properties of \( f(x) = e^x \)
- The function \( f(x) = e^x \) is defined for all real numbers \( x \).
- It is an exponential function that is always positive and increases rapidly as \( x \) increases.
- At \( x = 0 \), \( f(0) = e^0 = 1 \).
- As \( x \to -\infty \), \( f(x) \to 0 \) and as \( x \to +\infty \), \( f(x) \to +\infty \).

### Step 3: Properties of \( g(x) = \log |x| \)
- The function \( g(x) = \log |x| \) is defined for \( x \neq 0 \).
- For \( x > 0 \), \( g(x) = \log x \) which increases from \( -\infty \) as \( x \) approaches 0 to \( +\infty \) as \( x \) increases.
- For \( x < 0 \), \( g(x) = \log (-x) \) which behaves similarly, increasing from \( -\infty \) as \( x \) approaches 0 from the left to \( +\infty \) as \( x \) goes to negative infinity.

### Step 4: Finding intersections
To find the number of solutions, we need to find the points where \( f(x) \) intersects \( g(x) \).

1. For \( x > 0 \):
- As \( x \) approaches 0, \( g(x) \to -\infty \) while \( f(x) \to 1 \).
- As \( x \) increases, \( g(x) \) increases and eventually intersects \( f(x) \) at some point since \( f(x) \) continues to grow.

2. For \( x < 0 \):
- As \( x \) approaches 0 from the left, \( g(x) \to -\infty \) while \( f(x) \to 1 \).
- As \( x \) decreases, \( g(x) \) increases and will also intersect \( f(x) \) at some point.

### Step 5: Conclusion
From the analysis above, we can conclude that there is exactly one intersection point for \( x > 0 \) and one for \( x < 0 \). Therefore, the total number of solutions to the equation \( e^x - \log |x| = 0 \) is:

\[ \text{Number of solutions} = 1 + 1 = 2 \]

### Final Answer
The number of solutions of the equation \( e^x - \log |x| = 0 \) is **2**.

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