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

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Solution

second term is $^{5} C_{2} (x)^{2} (x^{log x})^{3} = 10^{6}$
$⟹ x^{2 + 3 log x} = 10^{5}$
$log x (3 log x + 2) = 5$
$3 (log x)^{2} + 2 log x - 5 = 0$
$3 (log x)^{2} + 5 log x - 3 log x - 5 = 0$
$log x = - \frac{3}{5}, 1$
$⟹ x = 10, 10^{- 3 / 5}$

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