# Solutions of a logarithmic equation.

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.

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 $$=4$$