If the third in the expansion of `[x + x^(logx)]^(6)` is ` 10^(6)` , then x (x>1) may be

Options

A. 1
B. 10
C. `10^(-5//2)`
D. 102

Answer

To solve the problem, we need to find the value of \( x \) such that the third term in the expansion of \( [x + x^{\log x}]^6 \) is equal to \( 10^6 \).

### Step-by-Step Solution:

1. **Identify the General Term in the Binomial Expansion**:
The general term \( T_{r+1} \) in the binomial expansion of \( (a + b)^n \) is given by:
\[ T_{r+1} = \binom{n}{r} a^{n-r} b^r \]
Here, \( a = x \), \( b = x^{\log x} \), and \( n = 6 \).

2. **Write the Third Term**:
The third term corresponds to \( r = 2 \):
\[ T_3 = \binom{6}{2} x^{6-2} (x^{\log x})^2 \]
Simplifying this, we get:
\[ T_3 = \binom{6}{2} x^4 (x^{\log x})^2 = \binom{6}{2} x^4 x^{2 \log x} \]

3. **Calculate the Binomial Coefficient**:
\[ \binom{6}{2} = \frac{6 \times 5}{2 \times 1} = 15 \]
Therefore, the third term becomes:
\[ T_3 = 15 x^4 x^{2 \log x} = 15 x^{4 + 2 \log x} \]

4. **Set the Third Term Equal to \( 10^6 \)**:
We are given that this term equals \( 10^6 \):
\[ 15 x^{4 + 2 \log x} = 10^6 \]

5. **Isolate the Exponential Term**:
Dividing both sides by 15:
\[ x^{4 + 2 \log x} = \frac{10^6}{15} \]
Simplifying the right side:
\[ \frac{10^6}{15} = \frac{10^6}{3 \times 5} = \frac{10^6}{15} = \frac{10^6}{15} \approx 6.6667 \times 10^5 \]

6. **Take Logarithm on Both Sides**:
Taking the logarithm of both sides:
\[ \log(x^{4 + 2 \log x}) = \log\left(\frac{10^6}{15}\right) \]
This simplifies to:
\[ (4 + 2 \log x) \log x = \log(10^6) - \log(15) \]
\[ (4 + 2 \log x) \log x = 6 - \log(15) \]

7. **Approximate \( \log(15) \)**:
Using \( \log(15) \approx 1.1761 \):
\[ (4 + 2 \log x) \log x \approx 6 - 1.1761 = 4.8239 \]

8. **Solve for \( x \)**:
This is a transcendental equation and can be solved by substituting values. We can start with \( x = 10 \):
\[ (4 + 2 \log(10)) \log(10) = (4 + 2 \cdot 1) \cdot 1 = 6 \]
Since \( 6 \) is close to \( 4.8239 \), \( x = 10 \) is a reasonable approximation.

### Final Answer:
Thus, \( x \) may be approximately equal to \( 10 \).