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This question already has an answer here:

How can I save these animations as gif? I need reeeally quick answer if possible: :D enter image description here

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marked as duplicate by Jens, Kuba, Oleksandr R., user9660, kale Jan 18 '16 at 22:17

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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    $\begingroup$ mathematica.stackexchange.com/q/27202/5478 $\endgroup$ – Kuba Jan 18 '16 at 18:06
  • $\begingroup$ Welcome to Mathematica.SE! I suggest the following: 1) As you receive help, try to give it too, by answering questions in your area of expertise. 2) Take the tour! 3) When you see good questions and answers, vote them up by clicking the gray triangles, because the credibility of the system is based on the reputation gained by users sharing their knowledge. Also, please remember to accept the answer, if any, that solves your problem, by clicking the checkmark sign! $\endgroup$ – Michael E2 Jan 18 '16 at 18:08
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    $\begingroup$ One way to attack this problem is to look up GIF in the documentation. $\endgroup$ – Michael E2 Jan 18 '16 at 18:09
  • $\begingroup$ Please let us know if the link given by @Kuba doesn't work for you. Otherwise, the question will be closed as a duplicate. $\endgroup$ – Jens Jan 18 '16 at 18:58
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Saved as a movie in mov format:

SetDirectory@NotebookDirectory[];

Figure1[c_] := Plot[c + Log[2, x], {x, -2, 2}];
{a, b} = {2, 3};

m1 = Manipulate[
   Show[Figure1[c], PlotRange -> {{-a, a}, {-b, b}}],
   {c, -1, 3, .0001}
   ];

Export["movie.mov", m1];

Figure2[c_] := Plot[c + Log[2, x + c], {x, -3, 3}];
{a, b} = {3, 3};

m2 = Manipulate[
   Show[Figure2[c], PlotRange -> {{-a, a}, {-b, b}}],
   {c, -1, 3, .0001}
   ];

Export["movie2.mov", m2];
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