Cannot connect to internet after a power cut yesterday

I work at a small company, we have set up our LAN using an ASUS RT-N18U. We are in a small office as part of a business park, our network comes in over wire, as part of a package supplied by the business park.

We had a power cut yesterday, and since then we cannot get to the internet. I can connect to our local server, printers, etc but that's it. I've no idea what the issue is, but if I could even get an idea if the problem is internal to our LAN or external that'd be a start

EDIT: I cannot ping googles (or any other) servers from the router console.

The router logs show the following:

Aug  1 02:00:11 syslogd started: BusyBox v1.17.4
Aug  1 02:00:11 kernel: klogd started: BusyBox v1.17.4 (2016-02-03 06:01:20 CST)
Aug  1 02:00:11 kernel: Linux version 2.6.36.4brcmarm (gitserv_asus@wireless-pub1) (gcc version 4.5.3 (Buildroot 2012.02) ) #1 PREEMPT Wed Feb 3 06:09:47 CST 2016
Aug  1 02:00:11 kernel: CPU: ARMv7 Processor [413fc090] revision 0 (ARMv7), cr=10c53c7f
Aug  1 02:00:11 kernel: CPU: VIPT nonaliasing data cache, VIPT nonaliasing instruction cache
Aug  1 02:00:11 kernel: Machine: Northstar Prototype
Aug  1 02:00:11 kernel: Ignoring unrecognised tag 0x00000000
Aug  1 02:00:11 kernel: Memory policy: ECC disabled, Data cache writeback
Aug  1 02:00:11 kernel: Built 1 zonelists in Zone order, mobility grouping on.  Total pages: 60416
Aug  1 02:00:11 kernel: Kernel command line: root=/dev/mtdblock2 console=ttyS0,115200 init=/sbin/preinit earlyprintk debug
Aug  1 02:00:11 kernel: Memory: 255708k/255708k available, 6436k reserved, 0K highmem
Aug  1 02:00:11 kernel: Virtual kernel memory layout:
Aug  1 02:00:11 kernel:     vector  : 0xffff0000 - 0xffff1000   (   4 kB)
Aug  1 02:00:11 kernel:     fixmap  : 0xfff00000 - 0xfffe0000   ( 896 kB)
Aug  1 02:00:11 kernel:     DMA     : 0xf7e00000 - 0xffe00000   ( 128 MB)
Aug  1 02:00:11 kernel:     vmalloc : 0xd0800000 - 0xf0000000   ( 504 MB)
Aug  1 02:00:11 kernel:     lowmem  : 0xc0000000 - 0xd0000000   ( 256 MB)
Aug  1 02:00:11 kernel:     modules : 0xbf000000 - 0xc0000000   (  16 MB)
Aug  1 02:00:11 kernel:       .init : 0xc0008000 - 0xc0037000   ( 188 kB)
Aug  1 02:00:11 kernel:       .text : 0xc0037000 - 0xc0390000   (3428 kB)
Aug  1 02:00:11 kernel:       .data : 0xc03a8000 - 0xc03c7ec0   ( 128 kB)
Aug  1 02:00:11 kernel: * Invalid signature of oopsbuf: FF-FF-FF-BE-FE-FF-FF-FF (len 4294967295)
Aug  1 02:00:11 kernel: External imprecise Data abort at addr=0x0, fsr=0x1c06 ignored.
Aug  1 02:00:11 kernel: Mount-cache hash table entries: 512
Aug  1 02:00:11 kernel: Found a AMD NAND flash:
Aug  1 02:00:11 kernel: Total size:  128MB
Aug  1 02:00:11 kernel: Block size:  128KB
Aug  1 02:00:11 kernel: Page Size:   2048B
Aug  1 02:00:11 kernel: OOB Size:    64B
Aug  1 02:00:11 kernel: Sector size: 512B
Aug  1 02:00:11 kernel: Spare size:  16B
Aug  1 02:00:11 kernel: ECC level:   8 (8-bit)
Aug  1 02:00:11 kernel: Device ID: 0x 1 0xf1 0x 0 0x1d 0x 1 0xf1
Aug  1 02:00:11 kernel: bio: create slab <bio-0> at 0
Aug  1 02:00:11 kernel: PCI: no core
Aug  1 02:00:12 kernel: PCI: no core
Aug  1 02:00:12 kernel: PCI: Fixing up bus 0
Aug  1 02:00:12 kernel: PCI: Fixing up bus 0
Aug  1 02:00:12 kernel: PCI: Fixing up bus 1
Aug  1 02:00:12 kernel: VFS: Disk quotas dquot_6.5.2
Aug  1 02:00:12 kernel: Dquot-cache hash table entries: 1024 (order 0, 4096 bytes)
Aug  1 02:00:12 kernel: pflash: found no supported devices
Aug  1 02:00:12 kernel: bcmsflash: found no supported devices
Aug  1 02:00:12 kernel: Boot partition size = 524288(0x80000)
Aug  1 02:00:12 kernel: lookup_nflash_rootfs_offset: offset = 0x200000
Aug  1 02:00:12 kernel: nflash: squash filesystem with lzma found at block 28
Aug  1 02:00:12 kernel: Creating 4 MTD partitions on "nflash":
Aug  1 02:00:12 kernel: 0x000000000000-0x000000080000 : "boot"
Aug  1 02:00:12 kernel: 0x000000080000-0x000000200000 : "nvram"
Aug  1 02:00:12 kernel: 0x000000200000-0x000004000000 : "linux"
Aug  1 02:00:12 kernel: 0x00000038d3e0-0x000004000000 : "rootfs"
Aug  1 02:00:12 stop_nat_rules: apply the redirect_rules!
Aug  1 02:00:12 kernel: Registering the dns_resolver key type
Aug  1 02:00:12 kernel: Spare area=64 eccbytes 56, ecc bytes located at:
Aug  1 02:00:12 kernel:  2 3 4 5 6 7 8 9 10 11 12 13 14 15 18 19 20 21 22 23 24 25 26 27 28 29 30 31 34 35 36 37 38 39 40 41 42 43 44 45 46 47 50 51 52 53 54 55 56 57 58 59 60 61 62 63
Aug  1 02:00:12 kernel: Available 7 bytes at (off,len):
Aug  1 02:00:12 kernel: (1,1) (16,2) (32,2) (48,2) (0,0) (0,0) (0,0) (0,0) 
Aug  1 02:00:12 kernel: Options: NO_AUTOINCR,NO_READRDY,BBT_SCAN2NDPAGE,
Aug  1 02:00:12 kernel: Creating 2 MTD partitions on "brcmnand":
Aug  1 02:00:12 kernel: 0x000004000000-0x000007ec0000 : "brcmnand"
Aug  1 02:00:12 kernel: 0x000007ec0000-0x000008000000 : "asus"
Aug  1 02:00:12 kernel: VFS: Mounted root (squashfs filesystem) readonly on device 31:3.
Aug  1 02:00:12 kernel: et: module license 'Proprietary' taints kernel.
Aug  1 02:00:12 kernel: Disabling lock debugging due to kernel taint
Aug  1 02:00:12 kernel: et_module_init: passivemode set to 0x0
Aug  1 02:00:12 kernel: et_module_init: txworkq set to 0x1
Aug  1 02:00:12 kernel: et_module_init: et_txq_thresh set to 0x400
Aug  1 02:00:12 kernel: eth0: Broadcom BCM47XX 10/100/1000 Mbps Ethernet Controller 6.37.14.105 (r485445)
Aug  1 02:00:12 kernel: wl_module_init: passivemode set to 0x0
Aug  1 02:00:12 WAN Connection: ISP's DHCP did not function properly.
Aug  1 02:00:12 kernel: wl_module_init: igs set to 0x0
Aug  1 02:00:12 kernel: wl_module_init: txworkq set to 0x1
Aug  1 02:00:12 kernel: eth1: Broadcom BCM4360 802.11 Wireless Controller 6.37.14.105 (r485445)
Aug  1 02:00:16 dnsmasq[382]: warning: interface ppp1* does not currently exist
Aug  1 02:00:16 RT-N18U: start httpd
Aug  1 02:00:17 syslog: Generating SSL certificate...
Aug  1 02:00:17 disk monitor: be idle
Aug  1 02:00:17 hour monitor: daemon is starting
Aug  1 02:00:17 hour monitor: daemon terminates
Aug  1 02:00:17 miniupnpd[427]: version 1.9 started
Aug  1 02:00:17 miniupnpd[427]: HTTP listening on port 54077
Aug  1 02:00:17 miniupnpd[427]: Listening for NAT-PMP/PCP traffic on port 5351
Aug  1 02:00:18 start_nat_rules: apply the nat_rules(/tmp/nat_rules_eth0_eth0)!
Aug  1 02:00:18 syslog: module ledtrig-usbdev not found in modules.dep
Aug  1 02:00:18 syslog: module leds-usb not found in modules.dep
Aug  1 02:00:18 kernel: SCSI subsystem initialized
Aug  1 02:00:20 kernel: xt_connbytes: Forcing CT accounting to be enabled
Aug  1 02:00:21 kernel: nf_conntrack_rtsp v0.6.21 loading
Aug  1 02:00:21 kernel: nf_nat_rtsp v0.6.21 loading
Aug  1 02:00:21 kernel: jnl: driver (lke_9.2.0, Feb  3 2016 06:59:53, LBD=ON) loaded at bf5fa000
Aug  1 02:00:21 wan: finish adding multi routes
Aug  1 02:00:22 kernel: ufsd:: trace mask set to 0000000f
Aug  1 02:00:22 kernel: ufsd: driver (lke_9.2.0 lke_9.2.0_r248656_b5, build_host("wireless-pub1"), acl, ioctl, bdi, sd2(0), wb, bz, tr, rsrc) loaded at bf607000
Aug  1 02:00:22 kernel: NTFS support included
Aug  1 02:00:22 kernel: Hfs+/HfsJ support included
Aug  1 02:00:22 kernel: optimized: speed
Aug  1 02:00:22 kernel: Build_for__ASUS_PRODUCTS_003_k2.6.36_2013-11-07_lke_9.2.0_r248656_b5
Aug  1 02:00:22 kernel: xhci_hcd 0000:00:0c.0: Failed to enable MSI-X
Aug  1 02:00:22 kernel: xhci_hcd 0000:00:0c.0: failed to allocate MSI entry
Aug  1 02:00:22 kernel: usb usb1: No SuperSpeed endpoint companion for config 1  interface 0 altsetting 0 ep 129: using minimum values
Aug  1 02:00:22 WAN Connection: WAN was restored.
Aug  1 02:00:48 hour monitor: daemon is starting
Aug  1 02:00:48 hour monitor: ntp is not syn
Aug  1 02:01:06 rc_service: udhcpc 432:notify_rc stop_upnp
Aug  1 02:01:06 rc_service: udhcpc 432:notify_rc start_upnp
Aug  1 02:01:06 rc_service: waitting "stop_upnp" via udhcpc ...
Aug  1 02:01:06 miniupnpd[427]: shutting down MiniUPnPd
Aug  1 02:01:08 miniupnpd[629]: version 1.9 started
Aug  1 02:01:08 miniupnpd[629]: HTTP listening on port 47237
Aug  1 02:01:08 miniupnpd[629]: Listening for NAT-PMP/PCP traffic on port 5351
Aug  1 02:01:09 kernel: HTB: quantum of class 10002 is big. Consider r2q change.
Aug  1 02:01:09 kernel: HTB: quantum of class 10060 is big. Consider r2q change.
Aug  1 02:01:09 dhcp client: bound 192.168.0.102 via 192.168.0.254 during 60 seconds.
Aug  1 02:01:17 hour monitor: daemon is starting
Aug  1 02:01:17 hour monitor: ntp is not syn
Aug  1 02:01:47 hour monitor: daemon is starting
Aug  1 02:01:47 hour monitor: ntp is not syn
Aug  1 02:02:17 hour monitor: daemon is starting
Aug  1 02:02:17 hour monitor: ntp is not syn
Aug  1 02:02:47 hour monitor: daemon is starting
Aug  1 02:02:47 hour monitor: ntp is not syn
Aug  1 02:03:17 hour monitor: daemon is starting
Aug  1 02:03:17 hour monitor: ntp is not syn

The interesting line from the log is this:

Aug  1 02:01:09 dhcp client: bound 192.168.0.102 via 192.168.0.254 during 60 seconds.

There are two interesting pieces of information on this line. First of all, 192.168.0.102 is a RFC 1918 address. Which is not what you should be expecting to see allocated to the WAN of your router. Moreover during 60 seconds is not exactly what I would expect it to say.

You haven't mentioned what equipment your WAN interface is connected to. But what you are seeing looks remarkably similar to what one would see on a router connected to a cable modem which has no connection on the cable side:

Apr  8 09:55:49 hostname dhclient: bound to 192.168.87.102 -- renewal in 26 seconds.

The similarity here is both are RFC 1918 addresses, and in both cases the lease time is much lower than normal. The low lease time is used by the modem to ensure that your router will quickly get a real IP address once connectivity has been restored.

So I would check the modem which has assigned that IP address to your router, because chances are that modem does not have internet connectivity. With a bit of luck you just need to restart that modem. If that does not help you can try restarting the router 30 minutes after you restarted the modem. If the connection still doesn't work you would have to contact the ISP responsible for the modem.