Solving a transcendental equation

[geoduck]

Suppose you have the equation

x^2*(log x)^{1/3}=-C

for very small C. I have a book claims that asymptotically, for very small C, the solution is:

x^2=C*\frac{2}{[logC]^{1/3} }

I'm not quite sure how to show this. If the 2 wasn't there, it looks like what they did was:

x^2=-\frac{C}{(log x)^{1/3}}

and just substituted -C for x on the RHS:

x^2=\frac{C}{(log C)^{1/3}}

I was thinking maybe you could substitute -C for x² instead to get:x^2=\frac{2^{1/3}C}{(log C)^{1/3}}

 

[JJacquelin]

I agree, there is a typo in the formula of the book : the power 1/3 is missing in the coefficient 2^(1/3)
An even better approximate can be derived (in attachment)