# Solve the equation $(\frac{1}{27}-3^{6-x^2})(\log_2{(4+5x}))=0$. Indicate the smallest root.

Solve the equation $$(\frac{1}{27}-3^{6-x^2})(\log_2{(4+5x}))=0$$. Indicate the smallest root.

Initially, my approach was to equate both parts to $$0$$. Solve equations and choose the smallest $$x$$. I got $$x=+3, x=-3$$ for the $$(\frac{1}{27}-3^{6-x^2})$$ part and $$-1.25$$ for the latter part. However, when I sketched the whole equation using graphic calculator, I saw that the $$x=-0.6$$ and $$3$$. Could you please explain, how do I solve the initial problem and why my solution is wrong ?

We want $$4+5x=\color{blue}1$$, hence $$x=-0.6$$.
Also, the domain is $$4+5x>0$$, hence we reject $$-3$$.