Well, if it was FB crashing you'd look for the FB crash log, not the Springboard. Even if FB took out the Springboard the original crash was from FB so that's the log you'd look at.
Sometimes it does happen so fast though that a log doesn't get written, or only partially written.
Here's a log from my most recent crash of FB. This was due to Copic, which had two updates that day (Friday the 8th). I know that because Copic was crashing when I was scrolling in AddressBook. Then I opened FB and it crashed right away. If you look at the first few lines of the log it's pretty evident that it crashed trying to access my contacts when the FB app was opening. Again, I don't read crash log, but some stuff jumps out at you when you put things in context.
Version:1.0 StartHTML:0000000149 EndHTML:0000005585 StartFragment:0000000199 EndFragment:0000005551 StartSelection:0000000199 EndSelection:0000005551 Incident Identifier: EEEAC4F9-7E2D-4C04-A32A-992221BCCDA4
CrashReporter Key: adb1cd2cfec7194db6c1a4ed65389ef1f5a89609
Report Type: 187
Sandbox Violation: Facebook deny file-read-metadata /private/var/mobile/Library/AddressBook/AddressBook.sqlitedb
Process: Facebook [537]
Path: /var/mobile/Applications/E814C6E2-DAEC-4086-93B4-74D0C30ADEE2/Facebook.app/Facebook
Load Address: 0xb2000
Identifier: Facebook
Version: ??? (???)
Code Type: arm (Native)
Parent Process: launchd [1]
Date/Time: 2013-03-08 23:24:46.697 -0700
OS Version: iOS 6.1 (10B143)
Report Version: 104
Thread 0:
0 libsystem_kernel.dylib 0x39346f28 lstat + 8
1 Foundation 0x319de7e1 _NSFileExistsAtPath + 181
2 Foundation 0x319e79d1 -[NSFileManager fileExistsAtPath:isDirectory:] + 117
3 xCon.dylib 0x025f5aef _logos_method$NSFileManager$NSFileManager$fileExistsAtPath$isDirectory$(NSFileManager*, objc_selector*, NSString*, signed char*) + 23
4 Copic.dylib 0x022c62f4
5 Copic.dylib 0x022c6588
6 CoreFoundation 0x3113e0a5 __CFNotificationCenterAddObserver_block_invoke_0 + 125
7 CoreFoundation 0x3113e1bb ____CFXNotificationPostToken_block_invoke_0 + 143
8 CoreFoundation 0x31146925 __CFRUNLOOP_IS_CALLING_OUT_TO_A_BLOCK__ + 13
9 CoreFoundation 0x31145ff9 __CFRunLoopDoBlocks + 161
10 CoreFoundation 0x31145253 __CFRunLoopRun + 1451
11 CoreFoundation 0x310b823d CFRunLoopRunSpecific + 357
12 CoreFoundation 0x310b80c9 CFRunLoopRunInMode + 105
13 GraphicsServices 0x34c9633b GSEventRunModal + 75
14 UIKit 0x32fd42b9 UIApplicationMain + 1121
15 Facebook 0x000b4bd5
16 Facebook 0x000b4b98
Binary Images:
0xb2000 - 0xb1affd Facebook armv7 <7cab8d7bacec344d970a03050649a1f3> /var/mobile/Applications/E814C6E2-DAEC-4086-93B4-74D0C30ADEE2/Facebook.app/Facebook
0x22b8000 - 0x22dcffb Copic.dylib armv6 <b444d22eb8b53203a24936ac275ef2fd> /Library/MobileSubstrate/DynamicLibraries/Copic.dylib
0x25f2000 - 0x25f7fff xCon.dylib armv7 <83b67a1c646f316d8bc3305de9fafd64> /Library/MobileSubstrate/DynamicLibraries/xCon.dylib
0x310af000 - 0x311e1ff1 CoreFoundation armv7s <bd8e6c9f94b43e3d9af96a0f03ff3011> /System/Library/Frameworks/CoreFoundation.framework/CoreFoundation
0x319d8000 - 0x31b9bffd Foundation armv7s <0f73c35ada563c0bb2ce402d282faefd> /System/Library/Frameworks/Foundation.framework/Foundation
0x32f7d000 - 0x334d0ff3 UIKit armv7s <62bee9294ca13738bd7ff14365dc8561> /System/Library/Frameworks/UIKit.framework/UIKit
0x34c91000 - 0x34c9cfff GraphicsServices armv7s <44b33c403523309c9e930818c7fced34> /System/Library/PrivateFrameworks/GraphicsServices.framework/GraphicsServices
0x39345000 - 0x3935bffd libsystem_kernel.dylib armv7s <79bea3ebfda132baba8f5b0ad6ab95f5> /usr/lib/system/libsystem_kernel.dylib