How can I find out when my Macbook Pro turned itself on?

I'm trying to track down a problem where my MPB occasionally turns itself on while still closed. Perhaps if I can identify the time it happens, it will reveal something useful. In the most recent case, I found it on at 7:25am, and the battery was at 73%, so I'd guess it hadn't been for more than an hour.

There's not much in system.log. At 4am these messages appear, but that's it:

Feb  6 04:02:08 Steves-MacBook-Pro-3 com.apple.xpc.launchd[1] (com.apple.sandboxd[6466]): Service exited due to SIGALRM | sent by kernel_task[0]
Feb  6 04:02:08 Steves-MacBook-Pro-3 syslogd[56]: ASL Sender Statistics
Feb  6 04:02:09 Steves-MacBook-Pro-3 Dropbox[892]: [0206/040209.126924:WARNING:dns_config_service_posix.cc(326)] Failed to read DnsConfig.
Feb  6 04:02:13 Steves-MacBook-Pro-3 com.apple.xpc.launchd[1] (com.apple.bsd.dirhelper[6657]): Endpoint has been activated through legacy launch(3) APIs. Please switch to XPC or bootstrap_check_in(): com.apple.bsd.dirhelper
Feb  6 07:25:40 localhost bootlog[0]: BOOT_TIME 1549398340 0

Where else can I look?

Update:

Edited section of pmset -g log:

2019-02-06 00:00:30 +1100 Assertions            PID 113(hidd) TurnedOn UserIsActive "com.apple.iohideventsystem.queue.tickle.4294970902.3" 00:00:00  id:0x0x900008dda [System: DeclUser kDisp]
2019-02-06 04:02:08 +1100 Kernel Client Acks    Delays to Sleep notifications: [AppleThunderboltNHIType2 driver is slow(msg: WillChangeState to 2)(366 ms)]
2019-02-06 04:02:08 +1100 Assertions            PID 71(powerd) Created InternalPreventSleep "Holding in darkwake for up to 30 seconds to update inactivity prediction" 00:00:00  id:0x0xd000093b4 [System: DeclUser SRPrevSleep kCPU kDisp]
2019-02-06 04:02:08 +1100 Assertions            PID 113(hidd) TurnedOn UserIsActive "com.apple.iohideventsystem.queue.tickle.4294970902.3" 00:00:00  id:0x0x900008dda [System: DeclUser SRPrevSleep kCPU kDisp]
2019-02-06 04:02:08 +1100 com.apple.sleepservices.sessionStarted    SleepService: window begins with cap time=180 secs
2019-02-06 04:02:08 +1100 DarkWake              DarkWake from Standby [CDNP] due to RTC/SleepService: Using BATT (Charge:97%)
2019-02-06 04:02:08 +1100 HibernateStats        hibmode=3 standbydelay=10800                                                          34
2019-02-06 04:02:08 +1100 WakeTime              WakeTime: 1.693 sec
2019-02-06 04:02:08 +1100 Kernel Client Acks    Delays to Wake notifications: [AppleIntelFramebuffer driver is slow(msg: SetState to 1)(495 ms)]
2019-02-06 04:02:08 +1100 Assertions            PID 183(mDNSResponder) Created MaintenanceWake "mDNSResponder:maintenance" 00:00:00  id:0x0xd000093ba [System: PrevIdle DeclUser PushSrvc SRPrevSleep kCPU kDisp]
2019-02-06 04:02:09 +1100 Assertions            PID 183(mDNSResponder) Released MaintenanceWake "mDNSResponder:maintenance" 00:00:00  id:0x0xd000093ba [System: PrevIdle DeclUser PushSrvc SRPrevSleep kCPU kDisp]
2019-02-06 04:02:09 +1100 Assertions            PID 113(hidd) TurnedOn UserIsActive "com.apple.iohideventsystem.queue.tickle.4294970902.3" 00:00:00  id:0x0x900008dda [System: DeclUser PushSrvc SRPrevSleep kCPU kDisp]
[repeats ~10 times]
2019-02-06 04:02:09 +1100 Notification          Next largest inactivity window start:'2019-02-07 00:00:00 +1100' end:'2019-02-07 04:00:00 +1100'
2019-02-06 04:02:09 +1100 Assertions            PID 71(powerd) Released InternalPreventSleep "Holding in darkwake for up to 30 seconds to update inactivity prediction" 00:00:00  id:0x0xd000093b4 [System: DeclUser PushSrvc kCPU kDisp]
[repeats ~165 times]

2019-02-06 07:25:47 +1100 HibernateStats        hibmode=0 standbydelay=0                                                              0
Sleep/Wakes since boot at 2019-02-04 19:28:51 +1100 :0   Dark Wake Count in this sleep cycle:3

You could try looking at the output of pmset -g log. You didn't specify your macOS version, but the last three OS versions (at least) have this utility.

It outputs a log of power-related events:

...
2019-01-30 02:21:22 -0800 DarkWake              DarkWake from Normal Sleep [CDN] due to RTC/SleepService: Using AC (Charge:100%) 45 secs   
2019-01-30 02:21:22 -0800 HibernateStats        hibmode=3 standbydelay=1800                                                           187           
2019-01-30 02:21:22 -0800 WakeTime              WakeTime: 1.799 sec  
...
2019-01-30 08:37:56 -0800 Assertions            PID 3724(PowerChime) Released DisplayWake "com.apple.chimeOnAttach" 00:00:04  id:0x0xe000090b9 [System: PrevIdle DeclUser BGTask SRPrevSleep IPushSrvc kCPU kDisp]          
Sleep/Wakes since boot:0   Dark Wake Count in this sleep cycle:7

Dark wakes are when the system wakes up to perform some limited activity. If you see lots and lots of dark wakes in a sleep cycle it might be somewhat problematic.

More often, it's something like a Bluetooth mouse or other network device being allowed to wake the system, which you might also see in the power log as a wake event.