Here's the shutdown cause:
Jan 12 00:58:48 localhost kernel[0]: Previous Shutdown Cause: 5
Here's the last bit of the log in case it is of some use. Like I said, the computer will shut down, without fail, at slightly over a minute of inactivity. If in continuous use, everything runs normally.
Jan 12 00:52:23 localhost kernel[0]: hi mem tramps at 0xffe00000
Jan 12 00:52:23 localhost kernel[0]: PAE enabled
Jan 12 00:52:23 localhost kernel[0]: 64 bit mode enabled
Jan 12 00:52:23 localhost kernel[0]: standard timeslicing quantum is 10000 us
Jan 12 00:52:23 localhost kernel[0]: vm_page_bootstrap: 250304 free pages
Jan 12 00:52:23 localhost kernel[0]: mig_table_max_displ = 71
Jan 12 00:52:23 localhost kernel[0]: Enabling XMM register save/restore and SSE/SSE2 opcodes
Jan 12 00:52:23 localhost kernel[0]: 93 prelinked modules
Jan 12 00:52:23 localhost kernel[0]: ACPI CA 20060421
Jan 12 00:52:23 localhost kernel[0]: AppleIntelCPUPowerManagement: ready
Jan 12 00:52:23 localhost kernel[0]: AppleACPICPU: ProcessorApicId=0 LocalApicId=0 Enabled
Jan 12 00:52:23 localhost kernel[0]: AppleACPICPU: ProcessorApicId=1 LocalApicId=1 Enabled
Jan 12 00:52:23 localhost kernel[0]: Copyright (c) 1982, 1986, 1989, 1991, 1993
Jan 12 00:52:23 localhost kernel[0]: The Regents of the University of California. All rights reserved.
Jan 12 00:52:23 localhost kernel[0]: using 5242 buffer headers and 4096 cluster IO buffer headers
Jan 12 00:52:23 localhost kernel[0]: Enabling XMM register save/restore and SSE/SSE2 opcodes
Jan 12 00:52:23 localhost kernel[0]: Started CPU 01
Jan 12 00:52:23 localhost kernel[0]: IOAPIC: Version 0x20 Vectors 64:87
Jan 12 00:52:23 localhost kernel[0]: ACPI: System State [S0 S3 S4 S5] (S3)
Jan 12 00:52:23 localhost kernel[0]: Security auditing service present
Jan 12 00:52:23 localhost kernel[0]: BSM auditing present
Jan 12 00:52:23 localhost kernel[0]: disabled
Jan 12 00:52:23 localhost kernel[0]: rooting via boot-uuid from /chosen: 2D2A2B8C-BE44-476A-BD90-80AF73E53AF0
Jan 12 00:52:23 localhost kernel[0]: Waiting on <dict ID="0"><key>IOProviderClass</key><string ID="1">IOResources</string><key>IOResourceMatch</key><string ID="2">boot-uuid-media</string></dict>
Jan 12 00:52:23 localhost kernel[0]: USB caused wake event (EHCI)
Jan 12 00:52:23 localhost kernel[0]: FireWire (OHCI) Lucent ID 5811 PCI now active, GUID 001b63fffe73a6ac; max speed s400.
Jan 12 00:52:23 localhost kernel[0]: Got boot device = IOService:/AppleACPIPlatformExpert/PCI0@0/AppleACPIPCI/SATA@1F,2/AppleAHCI/PRT2@2/IOAHCIDevice@0/AppleAHCIDiskDriver/IOAHCIBlockStorageDevice/IOBlockStorageDriver/FUJITSU MHW2080BHPL Media/IOGUIDPartitionScheme/Apple_HFS_Untitled_1@2
Jan 12 00:52:23 localhost kernel[0]: BSD root: disk0s2, major 14, minor 2
Jan 12 00:52:23 localhost kernel[0]: jnl: replay_journal: from: 6234112 to: 4074496 (joffset 0x254000)
Jan 12 00:52:23 localhost kernel[0]: CSRHIDTransitionDriver:

robe:
Jan 12 00:52:23 localhost kernel[0]: CSRHIDTransitionDriver::start before command
Jan 12 00:52:23 localhost kernel[0]: CSRHIDTransitionDriver::stop
Jan 12 00:52:23 localhost kernel[0]: IOBluetoothHCIController::start Idle Timer Stopped
Jan 12 00:52:23 localhost kernel[0]: Jettisoning kernel linker.
Jan 12 00:52:23 localhost kernel[0]: Resetting IOCatalogue.
Jan 12 00:52:23 localhost kernel[0]: GFX0: family specific matching fails
Jan 12 00:52:23 localhost kernel[0]: display: family specific matching fails
Jan 12 00:52:23 localhost kernel[0]: Matching service count = 2
Jan 12 00:52:23 localhost kernel[0]: Matching service count = 4
Jan 12 00:52:23 localhost kernel[0]: Matching service count = 4
Jan 12 00:52:23 localhost kernel[0]: Matching service count = 4
Jan 12 00:52:23 localhost kernel[0]: Matching service count = 4
Jan 12 00:52:23 localhost kernel[0]: Matching service count = 4
Jan 12 00:52:23 localhost kernel[0]: Previous Shutdown Cause: 5
Jan 12 00:52:23 localhost kernel[0]: GFX0: family specific matching fails
Jan 12 00:52:23 localhost kernel[0]: display: family specific matching fails
Jan 12 00:52:23 localhost kernel[0]: ath_attach: devid 0x24
Jan 12 00:52:23 localhost kernel[0]: Override HT40 CTL Powers. EEPROM Version is 14.4, Device Type 5
Jan 12 00:52:23 localhost kernel[0]: ath_descdma_setup: tx dd_desc_paddr = 0xe833000, length 0x46500(288000) bytes
Jan 12 00:52:23 localhost kernel[0]: ath_descdma_setup: beacon dd_desc_paddr = 0xa390000, length 0x90(144) bytes
Jan 12 00:52:23 localhost kernel[0]: mac 12.10 phy 8.1 radio 12.0
Jan 12 00:52:23 localhost kernel[0]: IPv6 packet filtering initialized, default to accept, logging disabled
Jan 12 00:52:23 localhost memberd[37]: memberd starting up
Jan 12 00:52:23 localhost mDNSResponder-108.6 (Jul 19 2007 11: 41:28)[30]: starting
Jan 12 00:52:23 localhost lookupd[42]: lookupd (version 369.6) starting - Tue Jan 12 00:52:23 2010
Jan 12 00:52:24 localhost diskarbitrationd[36]: disk0s2 hfs 72C688EB-6013-3852-9674-6B4FE2278138 Jonathan /
Jan 12 00:52:25 localhost kernel[0]: yukonosx: Ethernet address 00:17:f2:de:f2:77
Jan 12 00:52:25 localhost kernel[0]: AirPort_Athr5424ab: Ethernet address 00:1c:b3:b3:fc:00
Jan 12 00:52:26 localhost DirectoryService[41]: Launched version 2.1 (v353.6)
Jan 12 00:52:27 localhost kernel[0]: Registering For 802.11 Events
Jan 12 00:52:27 localhost kernel[0]: [HCIController][setupHardware] AFH Is Supported
Jan 12 00:52:27 Jonathans-Computer configd[34]: setting hostname to "Jonathans-Computer.local"
Jan 12 00:52:27 Jonathans-Computer lookupd[64]: lookupd (version 369.6) starting - Tue Jan 12 00:52:27 2010
Jan 12 00:52:28 Jonathans-Computer kernel[0]: display: Not usable
Jan 12 00:52:28 Jonathans-Computer /System/Library/CoreServices/loginwindow.app/Contents/MacOS/loginwindow: Login Window Application Started
Jan 12 00:52:28 Jonathans-Computer loginwindow[67]: Login Window Started Security Agent
Jan 12 00:52:29 Jonathans-Computer /System/Library/PrivateFrameworks/Apple80211.framework/Resources/airport: Currently connected to network Cassie
Jan 12 00:52:32 Jonathans-Computer mDNSResponder: Adding browse domain local.
Jan 12 00:52:32 Jonathans-Computer configd[34]: executing /System/Library/SystemConfiguration/Kicker.bundle/Contents/Resources/enable-network
Jan 12 00:52:32 Jonathans-Computer configd[34]: posting notification com.apple.system.config.network_change
Jan 12 00:52:32 Jonathans-Computer lookupd[104]: lookupd (version 369.6) starting - Tue Jan 12 00:52:32 2010
Jan 12 00:52:33 Jonathans-Computer sudo: root : TTY=unknown ; PWD=/ ; USER=root ; COMMAND=/sbin/kextload /System/Library/Extensions/speedit.kext
Jan 12 00:52:33 Jonathans-Computer kernel[0]: SpeedIt Kext is loading!
Jan 12 00:52:33 Jonathans-Computer kernel[0]: Supported states
Jan 12 00:52:33 Jonathans-Computer configd[34]: target=enable-network: disabled
Jan 12 00:58:48 localhost kernel[0]: hi mem tramps at 0xffe00000
Jan 12 00:58:48 localhost kernel[0]: PAE enabled
Jan 12 00:58:48 localhost kernel[0]: 64 bit mode enabled
Jan 12 00:58:48 localhost kernel[0]: standard timeslicing quantum is 10000 us
Jan 12 00:58:48 localhost kernel[0]: vm_page_bootstrap: 250304 free pages
Jan 12 00:58:48 localhost kernel[0]: mig_table_max_displ = 71
Jan 12 00:58:48 localhost kernel[0]: Enabling XMM register save/restore and SSE/SSE2 opcodes
Jan 12 00:58:48 localhost kernel[0]: 93 prelinked modules
Jan 12 00:58:48 localhost kernel[0]: ACPI CA 20060421
Jan 12 00:58:48 localhost kernel[0]: AppleIntelCPUPowerManagement: ready
Jan 12 00:58:48 localhost kernel[0]: AppleACPICPU: ProcessorApicId=0 LocalApicId=0 Enabled
Jan 12 00:58:48 localhost kernel[0]: AppleACPICPU: ProcessorApicId=1 LocalApicId=1 Enabled
Jan 12 00:58:48 localhost kernel[0]: Copyright (c) 1982, 1986, 1989, 1991, 1993
Jan 12 00:58:48 localhost kernel[0]: The Regents of the University of California. All rights reserved.
Jan 12 00:58:48 localhost kernel[0]: using 5242 buffer headers and 4096 cluster IO buffer headers
Jan 12 00:58:48 localhost kernel[0]: Enabling XMM register save/restore and SSE/SSE2 opcodes
Jan 12 00:58:48 localhost kernel[0]: Started CPU 01
Jan 12 00:58:48 localhost kernel[0]: IOAPIC: Version 0x20 Vectors 64:87
Jan 12 00:58:48 localhost kernel[0]: ACPI: System State [S0 S3 S4 S5] (S3)
Jan 12 00:58:48 localhost kernel[0]: Security auditing service present
Jan 12 00:58:48 localhost kernel[0]: BSM auditing present
Jan 12 00:58:48 localhost kernel[0]: disabled
Jan 12 00:58:48 localhost kernel[0]: rooting via boot-uuid from /chosen: 2D2A2B8C-BE44-476A-BD90-80AF73E53AF0
Jan 12 00:58:48 localhost kernel[0]: Waiting on <dict ID="0"><key>IOProviderClass</key><string ID="1">IOResources</string><key>IOResourceMatch</key><string ID="2">boot-uuid-media</string></dict>
Jan 12 00:58:48 localhost kernel[0]: USB caused wake event (EHCI)
Jan 12 00:58:48 localhost kernel[0]: FireWire (OHCI) Lucent ID 5811 PCI now active, GUID 001b63fffe73a6ac; max speed s400.
Jan 12 00:58:48 localhost kernel[0]: Got boot device = IOService:/AppleACPIPlatformExpert/PCI0@0/AppleACPIPCI/SATA@1F,2/AppleAHCI/PRT2@2/IOAHCIDevice@0/AppleAHCIDiskDriver/IOAHCIBlockStorageDevice/IOBlockStorageDriver/FUJITSU MHW2080BHPL Media/IOGUIDPartitionScheme/Apple_HFS_Untitled_1@2
Jan 12 00:58:48 localhost kernel[0]: BSD root: disk0s2, major 14, minor 2
Jan 12 00:58:48 localhost kernel[0]: jnl: replay_journal: from: 8286720 to: 4643328 (joffset 0x254000)
Jan 12 00:58:48 localhost kernel[0]: CSRHIDTransitionDriver:

robe:
Jan 12 00:58:48 localhost kernel[0]: CSRHIDTransitionDriver::start before command
Jan 12 00:58:48 localhost kernel[0]: CSRHIDTransitionDriver::stop
Jan 12 00:58:48 localhost kernel[0]: IOBluetoothHCIController::start Idle Timer Stopped
Jan 12 00:58:48 localhost kernel[0]: Jettisoning kernel linker.
Jan 12 00:58:48 localhost kernel[0]: Resetting IOCatalogue.
Jan 12 00:58:48 localhost kernel[0]: GFX0: family specific matching fails
Jan 12 00:58:48 localhost kernel[0]: display: family specific matching fails
Jan 12 00:58:48 localhost kernel[0]: Matching service count = 2
Jan 12 00:58:48 localhost kernel[0]: Matching service count = 4
Jan 12 00:58:48 localhost kernel[0]: Matching service count = 4
Jan 12 00:58:48 localhost kernel[0]: Matching service count = 4
Jan 12 00:58:48 localhost kernel[0]: Matching service count = 4
Jan 12 00:58:48 localhost kernel[0]: Matching service count = 4
Jan 12 00:58:48 localhost kernel[0]: Previous Shutdown Cause: 5
Jan 12 00:58:48 localhost kernel[0]: GFX0: family specific matching fails
Jan 12 00:58:48 localhost kernel[0]: display: family specific matching fails
Jan 12 00:58:48 localhost kernel[0]: ath_attach: devid 0x24
Jan 12 00:58:48 localhost kernel[0]: Override HT40 CTL Powers. EEPROM Version is 14.4, Device Type 5
Jan 12 00:58:48 localhost kernel[0]: ath_descdma_setup: tx dd_desc_paddr = 0xdbd9000, length 0x46500(288000) bytes
Jan 12 00:58:48 localhost kernel[0]: ath_descdma_setup: beacon dd_desc_paddr = 0x974a000, length 0x90(144) bytes
Jan 12 00:58:48 localhost kernel[0]: mac 12.10 phy 8.1 radio 12.0
Jan 12 00:58:48 localhost kernel[0]: IPv6 packet filtering initialized, default to accept, logging disabled
Jan 12 00:58:48 localhost memberd[37]: memberd starting up
Jan 12 00:58:48 localhost mDNSResponder-108.6 (Jul 19 2007 11: 41:28)[30]: starting
Jan 12 00:58:48 localhost lookupd[43]: lookupd (version 369.6) starting - Tue Jan 12 00:58:48 2010
Jan 12 00:58:49 localhost diskarbitrationd[36]: disk0s2 hfs 72C688EB-6013-3852-9674-6B4FE2278138 Jonathan /
Jan 12 00:58:50 localhost kernel[0]: yukonosx: Ethernet address 00:17:f2:de:f2:77
Jan 12 00:58:50 localhost kernel[0]: AirPort_Athr5424ab: Ethernet address 00:1c:b3:b3:fc:00
Jan 12 00:58:51 localhost DirectoryService[41]: Launched version 2.1 (v353.6)
Jan 12 00:58:52 localhost kernel[0]: Registering For 802.11 Events
Jan 12 00:58:52 localhost kernel[0]: [HCIController][setupHardware] AFH Is Supported
Jan 12 00:58:52 Jonathans-Computer configd[34]: setting hostname to "Jonathans-Computer.local"
Jan 12 00:58:52 Jonathans-Computer lookupd[64]: lookupd (version 369.6) starting - Tue Jan 12 00:58:52 2010
Jan 12 00:58:53 Jonathans-Computer kernel[0]: display: Not usable
Jan 12 00:58:53 Jonathans-Computer /System/Library/CoreServices/loginwindow.app/Contents/MacOS/loginwindow: Login Window Application Started
Jan 12 00:58:53 Jonathans-Computer loginwindow[67]: Login Window Started Security Agent
Jan 12 00:58:54 Jonathans-Computer /System/Library/PrivateFrameworks/Apple80211.framework/Resources/airport: Currently connected to network Cassie
Jan 12 00:58:55 Jonathans-Computer configd[34]: executing /System/Library/SystemConfiguration/Kicker.bundle/Contents/Resources/enable-network
Jan 12 00:58:55 Jonathans-Computer configd[34]: posting notification com.apple.system.config.network_change
Jan 12 00:58:55 Jonathans-Computer lookupd[102]: lookupd (version 369.6) starting - Tue Jan 12 00:58:55 2010
Jan 12 00:58:56 Jonathans-Computer sudo: root : TTY=unknown ; PWD=/ ; USER=root ; COMMAND=/sbin/kextload /System/Library/Extensions/speedit.kext
Jan 12 00:58:56 Jonathans-Computer kernel[0]: SpeedIt Kext is loading!
Jan 12 00:58:56 Jonathans-Computer kernel[0]: Supported states
Jan 12 00:58:56 Jonathans-Computer configd[34]: target=enable-network: disabled
Jan 12 00:58:57 Jonathans-Computer mDNSResponder: Adding browse domain local.