Question:
The value of \(x\) in the expression \((x + x^{\log_{10}x})^5\), if the third term in the expansion is 1,000,000, is
The value of \(x\) in the expression \((x + x^{\log_{10}x})^5\), if the third term in the expansion is 1,000,000, is
Show Hint
Take logarithm on both sides to solve exponential equations.
Updated On: Apr 23, 2026
- \(10, 10^{-3/2}\)
- \(100 \text{ or } 10^{-3/2}\)
- \(10 \text{ or } 10^{-5/2}\)
- None of these
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The Correct Option is C
Solution and Explanation
Step 1: Formula / Definition}
\[ T_3 = \binom{5}{2} x^{3} (x^{\log_{10}x})^2 = 10 \cdot x^{3 + 2\log_{10}x} \]
Step 2: Calculation / Simplification}
\[ 10 \cdot x^{3 + 2\log_{10}x} = 10^6 \]
\[ x^{3 + 2\log_{10}x} = 10^5 \]
Take \(\log_{10}\): \((3 + 2\log_{10}x)\log_{10}x = 5\)
Let \(y = \log_{10}x\): \(2y^2 + 3y - 5 = 0\)
\((2y + 5)(y - 1) = 0 \Rightarrow y = 1, y = -5/2\)
\(x = 10^1 = 10\) or \(x = 10^{-5/2}\)
Step 3: Final Answer
\[ x = 10 \text{ or } 10^{-5/2} \]
\[ T_3 = \binom{5}{2} x^{3} (x^{\log_{10}x})^2 = 10 \cdot x^{3 + 2\log_{10}x} \]
Step 2: Calculation / Simplification}
\[ 10 \cdot x^{3 + 2\log_{10}x} = 10^6 \]
\[ x^{3 + 2\log_{10}x} = 10^5 \]
Take \(\log_{10}\): \((3 + 2\log_{10}x)\log_{10}x = 5\)
Let \(y = \log_{10}x\): \(2y^2 + 3y - 5 = 0\)
\((2y + 5)(y - 1) = 0 \Rightarrow y = 1, y = -5/2\)
\(x = 10^1 = 10\) or \(x = 10^{-5/2}\)
Step 3: Final Answer
\[ x = 10 \text{ or } 10^{-5/2} \]
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