Google Cloud and Google.Cloud.Functions.Framework

I'm using Google Cloud function with .net core and Google.Cloud.Functions.Framework to log user activity. I've considered to use DB to store these activities, however at first step I will try to store it in simple text file on google storage (is it proper from architectural point of view?).

My question is how can I write to google bucket using .net core and google function (my function is triggered on HTTP) and looks simple like this

    public async Task HandleAsync(HttpContext context)
    {
         //code goes here
         await context.Response.WriteAsync("Activity logged");
    }