how to rearrange for y for: $\ln{|y|}= \frac{x^3}{3}$?

Question

Is it $y= \pm e^{\frac{x^3}{3}}$?

I am not sure about whether the plus minus sign is correct.

Just here to look for confirmation or correction! Thanks!

Answer 1

Hint: $$\log|y|=\log \sqrt{y^2}=\frac{1}{2}\log y^2$$ so the equation becomes $$\frac{1}{2}\log y^2=\frac{x^3}{3}$$ $$ y^2=e^{\frac{2}{3}x^3}$$ now we see the plus minus sign is necessary $$ y=\pm e^{\frac{1}{3}x^3}$$