The third term in the expansion of `(x + x ^[log_10 x])^5` is `10^6 `then x =

Options

A. `10^(-1)`
B. 10
C. `10^2`
D. `10^3`

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 Third Term in the Expansion**:
The general term in the binomial expansion of \( (a + b)^n \) is given by:
\[ T_k = \binom{n}{k-1} a^{n-(k-1)} b^{k-1} \]
For our case, \( a = x \), \( b = x^{\log_{10} x} \), and \( n = 5 \). The third term \( T_3 \) corresponds to \( k = 3 \):
\[ T_3 = \binom{5}{2} x^{5-2} (x^{\log_{10} x})^2 \]
Simplifying this gives:
\[ T_3 = 10 x^3 (x^{\log_{10} x})^2 = 10 x^3 x^{2 \log_{10} x} = 10 x^{3 + 2 \log_{10} x} \]

2. **Set the Third Term Equal to \( 10^6 \)**:
We know from the problem statement that:
\[ 10 x^{3 + 2 \log_{10} x} = 10^6 \]
Dividing both sides by 10:
\[ x^{3 + 2 \log_{10} x} = 10^5 \]

3. **Take Logarithm on Both Sides**:
Taking the logarithm base 10 on both sides:
\[ \log_{10}(x^{3 + 2 \log_{10} x}) = \log_{10}(10^5) \]
This simplifies to:
\[ (3 + 2 \log_{10} x) \log_{10} x = 5 \]

4. **Let \( \log_{10} x = T \)**:
Substituting \( T \) for \( \log_{10} x \):
\[ (3 + 2T) T = 5 \]
Expanding this gives:
\[ 2T^2 + 3T - 5 = 0 \]

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

6. **Convert Back to \( x \)**:
Recall that \( T = \log_{10} x \):
- If \( T = 1 \), then \( x = 10^1 = 10 \).
- If \( T = -\frac{5}{2} \), then \( x = 10^{-\frac{5}{2}} = \frac{1}{10^{2.5}} = \frac{1}{\sqrt{100000}} = 10^{-2.5} \).

### Final Solutions:
Thus, the possible values of \( x \) are:
- \( x = 10 \)
- \( x = 10^{-\frac{5}{2}} \)