If `x^(2"log"_(10)x) = 1000x`, then x equals to

Options

A. `10, sqrt(10)`
B. `10^(-1), 10sqrt(10)`
C. `10sqrt(10)`
D. `sqrt(10)`

Answer

To solve the equation \( x^{2 \log_{10} x} = 1000x \), we will follow these steps:

### Step 1: Take the logarithm of both sides
We start by taking the logarithm (base 10) of both sides of the equation:
\[ \log_{10}(x^{2 \log_{10} x}) = \log_{10}(1000x) \]

### Step 2: Apply the logarithmic properties
Using the property of logarithms that states \( \log(a^b) = b \log(a) \), we can simplify the left-hand side:
\[ 2 \log_{10} x \cdot \log_{10} x = \log_{10}(1000) + \log_{10}(x) \]
This simplifies to:
\[ 2 (\log_{10} x)^2 = \log_{10}(1000) + \log_{10}(x) \]

### Step 3: Simplify the right-hand side
We know that \( 1000 = 10^3 \), so:
\[ \log_{10}(1000) = 3 \]
Thus, the equation becomes:
\[ 2 (\log_{10} x)^2 = 3 + \log_{10} x \]

### Step 4: Rearrange the equation
Rearranging the equation gives us:
\[ 2 (\log_{10} x)^2 - \log_{10} x - 3 = 0 \]

### Step 5: Let \( y = \log_{10} x \)
Let \( y = \log_{10} x \). The equation now reads:
\[ 2y^2 - y - 3 = 0 \]

### Step 6: Factor the quadratic equation
We can factor the quadratic:
\[ (2y + 3)(y - 2) = 0 \]
Setting each factor to zero gives:
1. \( 2y + 3 = 0 \) → \( y = -\frac{3}{2} \)
2. \( y - 2 = 0 \) → \( y = 2 \)

### Step 7: Solve for \( x \)
Now we convert back to \( x \):
1. For \( y = -\frac{3}{2} \):
\[ \log_{10} x = -\frac{3}{2} \implies x = 10^{-\frac{3}{2}} = \frac{1}{10^{3/2}} = \frac{1}{\sqrt{1000}} = \frac{1}{31.6228} \approx 0.0316 \]
2. For \( y = 2 \):
\[ \log_{10} x = 2 \implies x = 10^2 = 100 \]

### Final Solutions
Thus, the solutions for \( x \) are:
\[ x = 10^{-\frac{3}{2}} \quad \text{and} \quad x = 100 \]