roots of $5^{(\log_5x)^2}+x^{\log_5x}=1250$

Question

If p and q are solution of the equation $5^{(\log_5x)^2}+x^{\log_5x}=1250$, then $\log_q(p)$ has the value....

By hit a trial i used $x=25$ and it is matching but not able do find it

Answer 1

Notice that

$$5^{(\log_5x)^2}=(5^{(\log_5x)})^{\log_5x}=x^{\log_5x}$$

So, our equation reduces to:

$$x^{\log_5x}=625$$

Now take $\log_5$ on both sides. Can you solve it from here now?

Answer 2

Hint:

Let $\log_5x=y\implies x=5^y$

$$5^{y^2}+(5^y)^{y}=1250\iff5^{y^2}=5^4$$

Can you take it home from here?