Install Log4net nuget package

Install-Package log4net

Update your bootstapper to:

public class Bootstrapper : DefaultNancyBootstrapper
{
    protected override void ApplicationStartup(TinyIoCContainer container, IPipelines pipelines)
    {
        base.ApplicationStartup(container, pipelines);
        XmlConfigurator.Configure();
    }

    protected override void RequestStartup(TinyIoCContainer container, IPipelines pipelines, NancyContext context)
    {
        base.RequestStartup(container, pipelines, context);
    }

    protected override void ConfigureApplicationContainer(TinyIoCContainer container)
    {
        base.ConfigureApplicationContainer(container);            
        container.Register(typeof(ILog), (c, o) => LogManager.GetLogger(typeof(Bootstrapper)));
    }
}

Ensure Log4net is in your web.config configSection

This should have already been done when the package installed, but double check.

<configSections>
    <section name="log4net" type="log4net.Config.Log4NetConfigurationSectionHandler, log4net" />
</configSections>