If the third term in the expansion of `(1/x + x^(log_(10) x) )^5` is 100, find x .

Answer

To solve the problem, we need to find the value of \( x \) such that the third term in the expansion of \( \left( \frac{1}{x} + x^{\log_{10} x} \right)^5 \) is equal to 100.

### Step-by-Step Solution:

1. **Identify the General Term**:
The general term \( T_r \) in the expansion of \( (a + b)^n \) is given by:
\[ T_{r+1} = \binom{n}{r} a^{n-r} b^r \]
Here, \( a = \frac{1}{x} \), \( b = x^{\log_{10} x} \), and \( n = 5 \).

2. **Find the Third Term**:
The third term \( T_3 \) corresponds to \( r = 2 \):
\[ T_3 = \binom{5}{2} \left( \frac{1}{x} \right)^{5-2} \left( x^{\log_{10} x} \right)^2 \]
Simplifying this:
\[ T_3 = \binom{5}{2} \left( \frac{1}{x} \right)^3 \left( x^{\log_{10} x} \right)^2 \]
\[ = 10 \cdot \frac{1}{x^3} \cdot x^{2 \log_{10} x} \]
\[ = 10 \cdot \frac{x^{2 \log_{10} x}}{x^3} \]
\[ = 10 \cdot x^{2 \log_{10} x - 3} \]

3. **Set the Third Term Equal to 100**:
We know from the problem statement that \( T_3 = 100 \):
\[ 10 \cdot x^{2 \log_{10} x - 3} = 100 \]
Dividing both sides by 10:
\[ x^{2 \log_{10} x - 3} = 10 \]

4. **Take Logarithm**:
Taking logarithm base 10 on both sides:
\[ \log_{10} \left( x^{2 \log_{10} x - 3} \right) = \log_{10} 10 \]
\[ (2 \log_{10} x - 3) \log_{10} x = 1 \]
Let \( \log_{10} x = \alpha \):
\[ (2\alpha - 3) \alpha = 1 \]
\[ 2\alpha^2 - 3\alpha - 1 = 0 \]

5. **Solve the Quadratic Equation**:
Using the quadratic formula \( \alpha = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = 2, b = -3, c = -1 \):
\[ \alpha = \frac{3 \pm \sqrt{(-3)^2 - 4 \cdot 2 \cdot (-1)}}{2 \cdot 2} \]
\[ = \frac{3 \pm \sqrt{9 + 8}}{4} \]
\[ = \frac{3 \pm \sqrt{17}}{4} \]

6. **Find Values of \( x \)**:
Since \( \alpha = \log_{10} x \):
\[ x = 10^{\alpha} = 10^{\frac{3 + \sqrt{17}}{4}} \quad \text{or} \quad x = 10^{\frac{3 - \sqrt{17}}{4}} \]

### Final Answer:
The possible values of \( x \) are:
\[ x = 10^{\frac{3 + \sqrt{17}}{4}} \quad \text{or} \quad x = 10^{\frac{3 - \sqrt{17}}{4}} \]