Amazon web services – Log an AWS APIGateway proxy transformation?

We have a lambda sitting inside of a private security group which needs to make a call to an outside API across the internet. To do this we decided to go with a proxy REST API that will allow the lambda to call the endpoint using an internal AWS address like this:

https://<myapigatewayid>.execute-api.<my-region>.amazonaws.com/<myname>/<my-proxy-call-path>/?arg1=123&arg2=345

The proxy integration looks like this:

    MyAPIGetProxyMethod:
      Type: AWS::ApiGateway::Method
      Properties:
        ResourceId: !Ref MyAPIGetProxyResource
        RestApiId: !Ref Proxy
        AuthorizationType: NONE
        HttpMethod: GET
        RequestParameters:
          method.request.querystring.arg1: true
          method.request.querystring.arg2: true
        Integration:
          IntegrationHttpMethod: GET
          Type: HTTP_PROXY
          RequestParameters:
            integration.request.header.Authorization: "'Bearer <my-bearer-token>'"
            integration.request.querystring.arg1: method.request.querystring.arg1
            integration.request.querystring.arg2: method.request.querystring.arg2
          Uri: https://some.api.com/reports/

We want to log the actual transformed URL that is being called in order to see if the format is correct. Ie something like this:

https://some.api.com/reports/?arg1=123&arg2=345

Q: does anyone know how to configure the APIGateway log for the API so that it will log the transformed URL? Currently our log format looks like this:

requestId: $context.requestId, ip: $context.identity.sourceIp, caller: $context.identity.caller, user: $context.identity.user, contextPath: $context.path, contextOverRidePath: $context.requestOverride.path.path_name, requestTime: $context.requestTime, httpMethod: $context.httpMethod, resourcePath: $context.resourcePath, status: $context.status, protocol: $context.protocol, responseLength: $context.responseLength