# Find the smallest value of $a$ that a solution for this equation exists?

How to find the smallest value of $a$ that a solution for this equation exists? $2\ln x + \frac{5}{x} < a$

That are the steps I've done, and have no idea what to do next:

$2\ln x < -\frac{5}{x} + a$

$\ln x^2 < -\frac{5}{x} + a$

$e^{-\frac{5}{x} + a} < x^2$

$e^{-5} \cdot e^{\frac{1}{x}} \cdot e^a < x^2$

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