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Equation from Question:

$$x^{(\log_3(x))^2 + \log_3(x^4)-3}=3^{2\log_3(x)}$$

The question states to find the number of solutions and the sum of integral solutions. The correct answer is 3 solutions of integral sum 4

Solution attempt

According to my solution, I only get two solutions out of which only one is integral (3). Please help me in finding the mistake.

PS: Sorry for not typing the equation but I don't know how to do so on phone! Also, Stack wouldn't let me post images because of repo.

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Let $t=\log_3 x$. Then,on taking $\log_3$ on both sides, we get - $$(t^2+4t-3)t=2t$$ Simplifying, we get $t=0,1,-5$, or $x=1,3,3^{\frac{1}{5}}$. Hence, sum of integral solutions $=4$

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  • $\begingroup$ Thanks! For the case when t=0! $\endgroup$ – Arnav Varshney May 19 at 8:48

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