The number of solution of ` x^( 1 + logx) = 10 x ` is __________

Answer

To solve the equation \( x^{1 + \log x} = 10x \), we will follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ x^{1 + \log x} = 10x \]

### Step 2: Take logarithm on both sides
Taking logarithm (base 10) on both sides, we have:
\[ \log(x^{1 + \log x}) = \log(10x) \]

### Step 3: Apply logarithmic properties
Using the property of logarithms, \( \log(a^b) = b \log a \), we can rewrite the left side:
\[ (1 + \log x) \log x = \log(10) + \log(x) \]

### Step 4: Simplify the right side
Since \( \log(10) = 1 \), we can rewrite the equation as:
\[ (1 + \log x) \log x = 1 + \log x \]

### Step 5: Rearrange the equation
Rearranging gives:
\[ (1 + \log x) \log x - (1 + \log x) = 0 \]
Factoring out \( (1 + \log x) \):
\[ (1 + \log x)(\log x - 1) = 0 \]

### Step 6: Set each factor to zero
Now we set each factor to zero:
1. \( 1 + \log x = 0 \)
2. \( \log x - 1 = 0 \)

### Step 7: Solve for \( x \)
From the first equation:
\[ \log x = -1 \]
This implies:
\[ x = 10^{-1} = 0.1 \]

From the second equation:
\[ \log x = 1 \]
This implies:
\[ x = 10^{1} = 10 \]

### Step 8: Count the solutions
The solutions we found are \( x = 0.1 \) and \( x = 10 \). Therefore, the number of solutions is:
\[ \text{Number of solutions} = 2 \]

### Final Answer
The number of solutions of the equation \( x^{1 + \log x} = 10x \) is **2**.

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