Changing the logo is a basic  when it comes to branding SharePoint 2013 sites.

If you have more site collection in SharePoint ,hundred of site collection with subsites , go with sharepoint Powershell script

Get-SPWebApplication "http://sharepoint.crescent.com" | Get-SPSite -Limit "All" | Get-SPWeb -Limit "All" |  foreach { $_.SiteLogoUrl=""; $_.update(); Write-host "Changing Logo for:"$_.Url;  }

Add-PSSnapin Microsoft.SharePoint.PowerShell -erroraction SilentlyContinue 

$SiteURL = "http://Your-Site-Collection-URL" 
#Get the site collection
$Site = Get-SPSite $SiteURL #Iterate through each site in the site collection
 foreach($web in $Site.AllWebs)
 {
 #sharepoint change logo programmatically 
$web.SiteLogoUrl = "/relative-path-logo-file" 
$web.Update() 
}

• SPListItem.Update() is used to update all the values in SharePoint List Item including Modified Date, Modified By, Version fields
• 
  
• SPListItem.SystemUpdate() is used to update the values of the list item without modifying the Modified Date, Modified By and Version fields.
• 
  
• SPListItem.Update() . Event Receiver is fired in case if listitemupdated,listitemupdating  event is associated. 
• SPListItem.SystemUpdate() Event Receiver won't be fired in case if list item updated, listitemupdating  event is associated

Hope It will works fine.