If the third term in the expansion of `[x+x^(log_(10)x)]^(5)` is `10^(6)`, then x may be

Options

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

Answer

To solve the problem, we need to find the value of \( x \) such that the third term in the expansion of \( (x + x^{\log_{10} x})^5 \) equals \( 10^6 \).

### Step-by-Step Solution:

1. **Identify the expression**:
We start with the expression \( (x + x^{\log_{10} x})^5 \).

2. **Define \( y \)**:
Let \( y = \log_{10} x \). Then, we can rewrite \( x^{\log_{10} x} \) as \( x^y \). Thus, the expression becomes:
\[ (x + x^y)^5 \]

3. **Find the third term**:
The third term in the binomial expansion of \( (a + b)^n \) is given by:
\[ T_3 = \binom{n}{r} a^{n-r} b^r \]
For our case, \( n = 5 \), \( a = x \), and \( b = x^y \). The third term corresponds to \( r = 2 \):
\[ T_3 = \binom{5}{2} x^{5-2} (x^y)^2 = \binom{5}{2} x^3 (x^{2y}) = \binom{5}{2} x^{3 + 2y} \]

4. **Calculate \( \binom{5}{2} \)**:
\[ \binom{5}{2} = \frac{5!}{2!(5-2)!} = \frac{5 \times 4}{2 \times 1} = 10 \]
Therefore, the third term becomes:
\[ T_3 = 10 x^{3 + 2y} \]

5. **Set the equation**:
We know that this term equals \( 10^6 \):
\[ 10 x^{3 + 2y} = 10^6 \]

6. **Divide both sides by 10**:
\[ x^{3 + 2y} = 10^5 \]

7. **Take logarithm base 10**:
Taking logarithm base 10 of both sides gives:
\[ \log_{10}(x^{3 + 2y}) = \log_{10}(10^5) \]
Using properties of logarithms:
\[ (3 + 2y) \log_{10} x = 5 \]

8. **Substitute \( y \)**:
Recall that \( y = \log_{10} x \):
\[ (3 + 2 \log_{10} x) \log_{10} x = 5 \]
Let \( z = \log_{10} x \):
\[ (3 + 2z) z = 5 \]
Expanding gives:
\[ 2z^2 + 3z - 5 = 0 \]

9. **Solve the quadratic equation**:
We can use the quadratic formula \( z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ z = \frac{-3 \pm \sqrt{3^2 - 4 \cdot 2 \cdot (-5)}}{2 \cdot 2} = \frac{-3 \pm \sqrt{9 + 40}}{4} = \frac{-3 \pm \sqrt{49}}{4} = \frac{-3 \pm 7}{4} \]
This gives us two potential solutions:
\[ z = \frac{4}{4} = 1 \quad \text{and} \quad z = \frac{-10}{4} = -2.5 \]

10. **Convert back to \( x \)**:
Recall \( z = \log_{10} x \):
- For \( z = 1 \):
\[ \log_{10} x = 1 \implies x = 10^1 = 10 \]
- For \( z = -2.5 \):
\[ \log_{10} x = -2.5 \implies x = 10^{-2.5} = \frac{1}{10^{2.5}} = \frac{1}{\sqrt{1000}} = \frac{1}{31.6228} \approx 0.0316 \]

### Final Answer:
Thus, the possible values of \( x \) are:
\[ x = 10 \quad \text{or} \quad x \approx 0.0316 \]