From what little I could make out (my eyes are terrible with a white background) it looks like you are running on a Mac. The log reports you have a total of 4GB of memory installed of which about 1.2GB is available. This is not enough Ram for Java/Minecraft to do much more than run Vanilla.
In order to run any but the lightest of 1.7 packs you will have to add some memory to your system. Judging from the amount of free ram listed in the log you need to load at least 2GB, 4GB would be better, since Java 1.8.0_xx has a default config as follows:
Minimum Ram 2GB
Maximum Ram 4GB
Perm. Gen. 256MB
Note: These amounts are required in addition to memory used by your O.S.