Struggles with "belongsTo()"

So I've got a Client model and a MaintenanceEntry model. I've got a pivot table setup to accommodate the relationship (a MaintenanceEntry belongsTo a single Client, but a Client hasMany MaintenanceEntry(s)). What I'm trying to do is fetch the Client id associated with an individual MaintenanceEntry, but when I try to call it, I just get null. This is how I'm calling it:

$maintenance_entry->client->id

but I've also tried

$maintenance_entry->client

//OR

$maintenance_entry->client()

//OR

$maintenance_entry->client()->id

and no luck! I feel like I'm having a hard time wrapping my mind around relationships. Fortunately whenever I've been able to trigger an error, the SQL errors were really useful in learning how to properly structure my databases to accommodate relationships. But like I said, it's just returning nul..

MaintenanceEntry Model:

<?php
namespace Cole;
use Illuminate\Database\Eloquent\Model;

class MaintenanceEntry extends Model{

    protected $table = "maintenance_entries";
    protected $fillable = ['type_id', 'notes'];

    public function type(){
        return $this->belongsTo('Cole\MaintenanceType');
    }

    public function client(){
        return $this->belongsTo('Cole\Client', 'client_id');
    }

}

Client Model:

<?php
namespace Cole;
use Illuminate\Database\Eloquent\Model;

class Client extends Model{

    protected $fillable = ['client','name','hosting','address','phone_1','phone_2','email_address'];

    public function contacts(){
        return $this->belongsToMany('Cole\Contact');
    }

    public function hosting(){
        return $this->belongsTo('Cole\Hosting');
    }

    public function maintenance(){
        return $this->belongsToMany('Cole\MaintenanceEntry');
    }

}

Then my DB is structured like this:

clients table: id (unsigned, primary, ai) name (varchar) website (varchar) hosting_id (int)

client_maintenance_entry table:

id (unsigned, primary, ai)

client_id (unsigned, FK->clients.id)

maintenance_entry_id(unsigned, FK->maintenance_entries.id)

maintenance_entries table:

id (usigned, primary, ai)

type_id

notes