Here is a sample I created that might help. Note: I use List types instead of ArrayList ones. Also, it assumes only one possible matching PC name in the data to be updated. You'll have to alter it to update the file since it merely updates the first List variable. Let me know how it goes.

Copy[PSCustomObject]
{
[string] $pc,
[string] $name,
[string] $date,
[string] $city
}

[System.Collections.Generic.List[PSCustomObject]] $list1 = Import-Csv "C:\SOSamples\log.csv";
[System.Collections.Generic.List[PSCustomObject]] $list2 = Import-Csv "C:\SOSamples\log2.csv";
[PSCustomObject] $record = $null;
[PSCustomObject] $match = $null;

foreach($record in $list2)
{
    # NOTE: This only retrieves the FIRST MATCHING item using a CASE-INSENSITIVE comparison       
    $match = $list1 | Where { $_.pc.ToLower() -eq $record.pc.ToLower() } | Select -First 1;

    if($match -eq $null)
    {
        Write-Host "Not Found!";
        $list1.Add($record);
    }
    else 
    {
        Write-Host "Found!";
        $match.date = $record.date;
    }
}

Write-Host "--------------------------------------------------------------------"

foreach($record in $list1)
{
    Write-Host $record
}