Question

Find $x$, if it is known that the second term of the expansion of $(x+x^{\log x}{)}^5$ is equal to $1000000$.

Answer 1

second term is ${}^5{C}_2\left(x{)}^2\right(x^{\log x}{)}^3={10}^6$

$⟹x^{2+3\log x}={10}^5$

$\log x(3\log x+2)=5$

$3(\log x{)}^2+2\log x-5=0$

$3(\log x{)}^2+5\log x-3\log x-5=0$

$\log x=-\frac{3}{5},1$

$⟹x=10,{10}^{-3/5}$