I took another look , starting with Log::Log4perl::Appender::String docs was a mistake (misleading), should have looked only at Log::Log4perl::Appender

Here it is, with layout, you should probably run this by the log4perl support list for commentary

#!/usr/bin/perl --
use strict; use warnings;
use Tkx;
use Log::Log4perl; Log::Log4perl->easy_init( $Log::Log4perl::INFO  );


Main( @ARGV );
exit( 0 );


sub Main {
#~     $Tkx::TRACE=64;
    Tkx::package_require('tile');
    Tkx::ttk__setTheme('xpnative');#Tkx::tile__setTheme('xpnative');
    
    my $logger = Log::Log4perl->get_logger; ## jellisii2
    my $appender = Log::Log4perl::Appender->new(
        "Log::Log4perl::Appender::String2",
    );
    $appender->layout( ## jellisii2
        Log::Log4perl::Layout::PatternLayout->new( "%d %m%n" ),
    );
    $logger->add_appender(  $appender   ); ## jellisii2
    
    
    my $mw = Tkx::widget->new(".");


    my $eee = $mw->new_entry(
        -textvariable => $appender->string_ref,  ## jellisii2
    );
    $eee->g_pack( -fill => "x", -expand => 1 );


    my $t = $mw->new_text(
        -padx => 5,
        -pady => 5,
        -background => "white",
    );
    $t->g_pack;
    
    $appender->tktext( $t ); ## jellisii2
    
    
    my $b = $mw->new_ttk__button(
        -text => 'Hello, world',
        -command => sub {
            $logger->info("hi"); ## jellisii2
        },
    );
    $b->g_pack;
    
    $b->configure();
    
    $logger->info("it begins"); ## jellisii2
    
    Tkx::MainLoop();
    
}

BEGIN {
    $INC{'Log/Log4perl/Appender/String2.pm'} = __FILE__;
    package Log::Log4perl::Appender::String2;
    use Log::Log4perl::Appender;
    our @ISA = qw(Log::Log4perl::Appender);
    
    sub new {
        my $proto  = shift;
        my $class  = ref $proto || $proto;
        my %params = @_;

        my $self = {
            name      => "unknown name",
            string    => "",
            %params,
        };

        bless $self, $class;
    }

    sub log {
        my( $self , %params ) = @_;
        
        my $message = $params{message} ;
        
        return unless $message;
        
        if( my $status_text = $self->tktext ){
            # text control has to be enabled to write to it.  
            $status_text->configure(-state => "normal");

            $status_text->insert(
                "end",
                $message
            );
            $status_text->see("end");

            # Don't want the user editing this...
            $status_text->configure(-state => "disabled");
            eval { $status_text->update(); 1} or eval { Tkx::update();
+ };
        }
        
        $self->{string} = "($message)";
        
    }
    
    sub string { # getter/setter
        my($self, $new) = @_;

        if(defined $new) {
            $self->{string} = $new;
        }

        return $self->{string};
    }
    
    sub tktext { # getter/setter
        my($self, $new) = @_;
        if(defined $new) {
            $self->{tktext} = $new;
        }

        return $self->{tktext};
    }

    sub string_ref { # getter
        my($self ) = @_;
        return \$self->{string};
    }
    1;

}
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I owe you a $BEVERAGE. Or perhaps several. Thank you very much.