I think the current answer just creates a GIF but doesn't explain why LocalizeVariables -> False would cause an abnormal GIF to be exported. I don't see any explanation about this in the documentation. I don't sure this is a bug or not. So I hope to reopen this post.

Code:

g1 = Manipulate[Plot[{Log[x], (x^(1-a)-1)/(1-a)}, {x, 0, 5}, PlotLegends->"Expressions"], 
{a, 0, 0.99}, LocalizeVariables->False];

Export["isoelastic.gif",g1]

Output (GIF):

As you can see, the .gif is just a static image. Any ideas?

EDIT:

But this code worked perfectly fine:

a1 = Manipulate[Plot[{(1-t)*x - t*((E^(-t*x)) - 1)}, {x, -100, 100}], {t, 0, 1}]

Export["graph.gif", a1]

Output: