fwrite() Isn't creating nor writing to a file that I specify it to

Question

"LoggedIn" does = true and "CurrentUser" isn't "ManselD", yet it's still not writing or creating the file :( And yes my /accounts/logs folder is CHMOD 702

if(!$user == "ManselD"){
    ini_set('date.timezone', 'Europe/London');
    $ip = $_SERVER['REMOTE_ADDR'];
    $txt = "$user Logged In With The Ip: $ip At ".date("h:i")."\n";
    $url = "/accounts/logs/".trim(date('F')."-".date('d')."-".date('Y').".txt");
    $file = fopen($url, "a");
    fwrite($file, $txt);
    fclose($file);
    echo $file;
    if(is_writable("/accounts/logs")){
        echo "It is writable";
    } else{
        echo "It isn't writable";
    }
}

I'm literally stumped and baffled at why this doesn't work :S

FIXED VERSION:

if($user != "ManselD"){
ini_set('date.timezone', 'Europe/London');
$ip = $_SERVER['REMOTE_ADDR'];
$txt = "$user Logged In With The Ip: $ip At ".date("h:i")."\n";
$url = getcwd() . '/accounts/logs/'.trim(date('F')."-".date('d')."-".date('Y').".txt");
file_put_contents($url, $txt, FILE_APPEND);
}

Answer 1

I think the problem (that you are asking about, because there are more if you read the comments) is that you write

$url = "/accounts/logs/".trim(date('F')."-".date('d')."-".date('Y').".txt");

The file paths are not URLs. And while the above is obviously not a URL, it should probably not have the starting slash (/).
A good approach would be to always use absolute paths, possibly using the __FILE__ constant.