conf t
logging monitor debugging
end

2. Turn on display of logging for current vty session and next turn on ip routing table debugging

terminal monitor
debug ip routing

3. To turn off display of logging for current vty session and turn off debugging

terminal no monitor
no debug ip routing

example:

Router(config)#ip route 0.0.0.0 0.0.0.0 192.168.0.1 name DEFAULT-GW
Router(config)#
*Oct 21 06:04:07.822: RT: add 0.0.0.0/0 via 192.168.0.1, static metric [1/0]
*Oct 21 06:04:07.822: RT: NET-RED 0.0.0.0/0
*Oct 21 06:04:07.822: RT: default path is now 0.0.0.0 via 192.168.0.1
*Oct 21 06:04:07.822: RT: new default network 0.0.0.0
*Oct 21 06:04:07.822: RT: NET-RED 0.0.0.0/0
Router(config)#no ip route 0.0.0.0 0.0.0.0 192.168.0.1
Router(config)#
*Oct 21 06:04:20.810: RT: del 0.0.0.0 via 192.168.0.1, static metric [1/0]
*Oct 21 06:04:20.810: RT: delete network route to 0.0.0.0
*Oct 21 06:04:20.810: RT: NET-RED 0.0.0.0/0
*Oct 21 06:04:20.810: RT: NET-RED 0.0.0.0/0