The errors gives us some hint about what wrong happening as follows :
[Wed Mar 07 03:02:27.719608 2018] [:error] [pid 21555] [client 108.162.250.6:36139]   File "/usr/local/lib/python2.7/dist-packages/selenium/webdriver/firefox/webdriver.py", line 151, in __init__
[Wed Mar 07 03:02:27.719611 2018] [:error] [pid 21555] [client 108.162.250.6:36139]     log_path=log_path)
As per the source code the GeckoDriver gets initiated with two default arguments executable_path and log_path=log_path as follows :
    if capabilities.get("marionette"):
        capabilities.pop("marionette")
        self.service = Service(executable_path, log_path=log_path)
        self.service.start()
Your program errors out here as the Value log_path (the log_file) corresponding to Key log_path is not editable (appendable) which finally fails in :
[Wed Mar 07 03:02:27.719617 2018] [:error] [pid 21555] [client 108.162.250.6:36139]     log_file = open(log_path, "a+") if log_path is not None and log_path != "" else None
[Wed Mar 07 03:02:27.719620 2018] [:error] [pid 21555] [client 108.162.250.6:36139] IOError: [Errno 13] Permission denied: 'geckodriver.log'
As per the source code, GeckoDriver Service by default is started as follows :
class Service(service.Service):
    """Object that manages the starting and stopping of the
    GeckoDriver."""
def __init__(self, executable_path, port=0, service_args=None,
             log_path="geckodriver.log", env=None):
    """Creates a new instance of the GeckoDriver remote service proxy.
    GeckoDriver provides a HTTP interface speaking the W3C WebDriver
    protocol to Marionette.
    :param log_path: Optional path for the GeckoDriver to log to.
        Defaults to _geckodriver.log_ in the current working directory.
    """
    log_file = open(log_path, "a+") if log_path is not None and log_path != "" else None
Which implies that if you do not pass the location of geckodriver.log explicitly through your program, GeckoDriver tends to create a file on it's own in the current working directory and in absence of the required permission it errors out with the message Permission denied: 'geckodriver.log'
Solution
The first and foremost point will be to check the compatibility between the binaries you are using.
A solution would be to :
- Initialize the GeckoDriver with the required arguments - executable_pathand- log_pathwith effective values(chmod 777- geckodriver.log) as follows :
 - def get_index(api_key):
    if str(api_key)!=the_api_key:
    return 401
    base_url = 'www.google.com'
    browser = webdriver.Firefox(executable_path="/usr/local/bin/geckodriver", log_path="/path/to/geckodriver.log")
    browser.get(base_url)
    html = browser.page_source
    return html         
 
- If you intend to create the - geckodriver.logwithin the Project Workspace ensure the required permission (chmod 777- Project Workspace) as follows :
 - def get_index(api_key):
    if str(api_key)!=the_api_key:
    return 401
    base_url = 'www.google.com'
    browser = webdriver.Firefox(executable_path='/usr/local/bin/geckodriver')
    browser.get(base_url)
    html = browser.page_source
    return html