An algebra problem by Aira Thalca

Algebra Level 3

\[\large 25^x = 10^x + 4^{x+1}\]

The solution \(x\) of the equation above can be written as \(\log_{\frac{a}{b}}\left(\frac{c+\sqrt{d}}e\right)\), where \(a\), \(b\), \(c\), \(d\) and \(e\) are positive integers with \(a\) and \(b\) being relatively primes and \(d\) square-free.

Find the value of the product \(abcde\).

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