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Q.
The value of $x$ in the expression $\left(x+x^{\log _{10} x}\right)^{5}$ if the third term in the expansion is 1000000 is
Binomial Theorem
Solution:
$\log x$ is defined only when $x>0$
Now, the 3rd term in the expansion is $1000000$
$\Rightarrow { }^{5} C_{2} x^{5-2} \cdot\left(x^{\log _{10} x}\right)^{2}=1000000$
$\Rightarrow x^{3+2 \log _{10} x}=10^{5}$
Taking logarithm of both sides, we get
$\left(3+2 \log _{10} x\right) \log _{10} x=5$
$\Rightarrow 2 y^{2}+3 y-5=0,$ where $\log _{10} x=y$
$\Rightarrow (y-1)(2 y+5)=0$
$\Rightarrow y=1$ or $-5 / 2$
$\Rightarrow \log _{10} x=1$ or $-5 / 2$
$\Rightarrow x=10^{1}=10$ or $10^{-5 / 2}$