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My attempt

$\log_5(4x^{log_5x}+5)=2(log_5x)^2$ Using $\log_ab=\frac{\log_cb}{\log_ca}$

$4x^{log_5x}+5=5^{2(log_5x)^2}$

Let $f=x^{log_5x}\Rightarrow \log_5 f = (log_5x)^2 \Rightarrow f= 5^{(\log_5 x)^2} $

We arrive at,

$ 4 *5^{(\log_5 x)^2}+5=5^{2(log_5x)^2};$ Let $5^{(\log_5 x)^2} = z$

$z^2-4z-5=0$ It gives;

$z=-1, 5$

$z=5$ gives $x=5, \frac{1}{5}.$ [Real solution]

$z=-1$ gives $5^{\sqrt{\log_5 (-1)}} = -0.0941629 - 4.90284 I$ [Complex solution]

Ques: I am suspecting that this complex solution should not be there, Could you please help me ?

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It is so simple, check the domain and range of $$5^{(\log_5 x)^2}$$

We know that $$(\log_5 x)^2\ge0$$ So, we get $$5^{(\log_5 x)^2}\ge1$$ or $$z\ge 1$$

So, $$z=5 $$ is the only solution.

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  • $\begingroup$ oh!, I missed this point. thank you very much $\endgroup$ – Epsilon Jun 2 at 7:23

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