MacBook can't reinstall OS X

I have been given a Macbook A1181 to repair by a friend. I am mainly a PC man but have mended a few macs over the years so I thought I could do it.

The problem was every time she started it up and tried to do anything it would either take ages e.g. twenty mins to open itunes or you would just get the nice spinny coloured wheel and nothing would happen. I booted into safe mode and exactly the same thing happened. This problem had only occurred after she had updated it to OS X Mavericks

I checked the disk for bad sectors using a program on my PC and it found and repaired one, seeing as it was just one sector I figured it didn’t need replacing

So I booted into the recovery console and did the repair disk, but it found no errors, so next I tried the reinstall OS X, everything seemed to be going OK, it downloaded from the internet OK & rebooted I selected Macintosh HD says its installing then I get
“The OS X installation couldn’t be completed
Blah blah blah blah about saving the log file”

Originally it wouldn’t let me save it anywhere, not to the HDD or memory stick, but now I have just managed to save it, so I will attach that, the formatting is all over the place if opened in Notepad but in Word it looks OK, hopefully someone can make sense of it.

Using HFSExplorer program on my PC I have managed to backup all her files, so if needs be the hard drive should be ok to wipe now

Just for info
3GB RAM
160GB HDD, 110GB partition Macintosh HD, 40GB BOOTCAMP & 600MB Recovery partition

Any ideas what the problem is and how to get round it?

Thanks,
Steve

It turns out she has the OSX 10.5.6 install DVD so I've started up from that and wiped all the partitions and it seems to be going oK so far.

Thanks

I deleted the two partitions, then created just one new partition in Disk Utility and it has installed OK.
I have updated it to 10.5.8 but now just wondering how to try updating to Mavericks again

I have updated it to 10.6.8 but when I tried to update to 10.9 from USB it booted Ok then displayed "there was a problem installing mac os x try reinstalling"

Would I just be better off updating to Yosemite? She says she doesnt use any old apps or programs, just ones she has got from the app store so I dont think there should be a problem there but will it run slower on this machine?

2.13GHz Core 2 Duo
3GB RAM - might be able to get it to 4GB as I read Mavericks really needs at least 4

MODEL identifer macbook5,2

OK I've got Yosemite downloading from the App Store now

Thanks for all your help, much appreciated!!

If those ID's are important then you had better remove them from your post as well I guess