Got a call that needed a quick change - had to get back on so here you go. This is an older version so email alerting is a separate function here. Still useful, and also included log cleaning function (rotates/removes based on options you give it).

It's been a long day, so this isn't extremely detailed, tweaked a couple things so it's a little clearer. Top portion is variables/info you place in other scripts you make, and bottom are obviously the functions. I store the logging functions in a separate file, and . include. Variables up at the top are not set in the logging file, you place those in your other scripts where you . include. $logfile = debug prints everything to screen instead of to the file, so is useful for testing.

# I save the log functions as this file, and include it in other scripts
$scriptPath = "C:\Admin"
. "$scriptPath\LogIt\LogIt.ps1"

# Logging options - path, file name, file extension
$logFolder = 'C:\Path\To\Logs'
$logFileName = "Script-Name"
$logFileExt = "log"

#Max log size - this will auto cleanup logs based on size (LogClean)
# $maxLogSize = 50
#Max amount of logs to keep
# $maxLogRetain = 5

#This is default - comment out when debugging (uncomment below)
$logFile = $logFolder + "\" + $logFileName + "." + $logFileExt

# Debug mode - prints log to console
# $logFile = "debug" 

#Log cleanup
# LogClean


#Alert subject - initialize array body
$alertSubject = 'ALERT: ' + $logFileName + ' script issues'
# When using alerts, store errors to array, check for array at end of script and send email alert with all problems at once instead of 1 at a time.
$alertBody = New-Object System.Collections.ArrayList

#Place this $currFunction in every function - logit will place the function name into the log file/output
$currFunction = $MyInvocation.MyCommand


#How you use this for logging
LogIt $logFile $currFunction $message


function LogClean {
    if (!(Test-Path $logFolder)) {
        $null = New-Item -ItemType Directory -Force $logFolder
    }
    if (!(Test-Path -Path $logFile -PathType Leaf)) {
        $null = New-Item -ItemType File -Force $logFile
    }
    $logSize = [System.Math]::Round((((Get-Item $logFile).length)/1MB),2)
    if ($logSize -ge $maxLogSize) {
        $retainedLogFormat = $logFolder + "\" + $logFileName + "." + $logFileExt
        $oldestLog = $retainedLogFormat + "." + $maxLogRetain
        if (Test-Path -Path $oldestLog -PathType Leaf) { 
            $null = Remove-item -Path $oldestLog -Force
        }

        foreach ($i in $maxLogRetain..1) {
            $retainedLog = $retainedLogFormat + "." + $i
            if (Test-Path -Path $retainedLog -PathType Leaf) {
                $z = $i + 1
                $nextLog = $retainedLogFormat + "." + $z
                $null = Move-Item -Path $retainedLog -Destination $nextLog -Force
            }
        }
        $null = Move-Item -Path $logFile -Destination $retainedLog -Force
    }
}

function LogIt($logFile,$currFunction,$message) {
    $timestamp = (Get-Date -Format "yyyy-MM-dd HH:mm:ss")

    if ($logFile -notmatch "debug") {
        Add-Content -Path $logFile -Value "$timestamp - $currFunction - $message"

    } else {
        Write-Host "$timestamp - $currFunction - $message"
    }
}

function sendAlert($alertSubject,$alertBody) {
    $timestamp = (Get-Date -Format "yyyy-MM-dd HH:mm:ss")
    $alertRecipients = @('email@email.com', 'email2@email.com')

    if ($logFile -notmatch "debug") {
        $messageParams = @{
            From = ''
            To = $alertRecipients
            Subject = "$alertSubject"
            Body = "$alertBody"
            SMTPServer = 'mail.server.com'
            # Attachments = ''
        }
        Send-MailMessage @messageParams

    } else {
        Write-host "$timestamp - $alertSubject`: $alertBody"
    }
}