Where can I log an ASP.NET Core app’s start/stop/error events?

Note: For .NET Core 3.0 or later, this answer is obsolete. See this answer instead.

You need to use Microsoft.AspNetCore.Hosting.IApplicationLifetime

    /// <summary>
    /// Triggered when the application host has fully started and is about to wait
    /// for a graceful shutdown.
    /// </summary>
    CancellationToken ApplicationStarted { get; }

    /// <summary>
    /// Triggered when the application host is performing a graceful shutdown.
    /// Requests may still be in flight. Shutdown will block until this event completes.
    /// </summary>
    CancellationToken ApplicationStopping { get; }

    /// <summary>
    /// Triggered when the application host is performing a graceful shutdown.
    /// All requests should be complete at this point. Shutdown will block
    /// until this event completes.
    /// </summary>
    CancellationToken ApplicationStopped { get; }

Instance of IApplicationLifetime could be obtained in Configure method. Also add ILoggerFactory here:

public void Configure(IApplicationBuilder app, IApplicationLifetime applicationLifetime, ILoggerFactory loggerFactory)
{
    // use applicationLifetime
}

Having ILoggerFactory, you can create instance of ILogger:

var logger = loggerFactory.CreateLogger("StartupLogger"); 

So you just need to create a property in the Startup class to persist the instance of ILogger (or ILoggerFactory, if you would like to create different ligger instance for different events). To summarize:

public class Startup 
{
    private ILogger _logger;

    public void Configure(IApplicationBuilder app, IApplicationLifetime applicationLifetime, ILoggerFactory loggerFactory) 
    {
        applicationLifetime.ApplicationStopping.Register(OnShutdown);
        ... 
        // add logger providers
        // loggerFactory.AddConsole()
        ...
        _logger = loggerFactory.CreateLogger("StartupLogger");
    }

    private void OnShutdown()
    {
         // use _logger here;
    }
}