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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 ?

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We want $4+5x=\color{blue}1$, hence $x=-0.6$.

Also, the domain is $4+5x>0$, hence we reject $-3$.

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