PHP (>=5.4), 90
#PHP (>=5.4), 90 (-2 for an anoynomous function in PHP 7; 74 for a snippet)
function f($s){$b+=7*(8>$b=ord(max(str_split($s)))-54);return[base_convert($s,$b,10),$b];}
Thanks @user34736 for the -54(+8) idea; saved two bytes on my -54/-47 approach.
The function returns an array instead of printing the result. +/-0 for printing:
function f($s){$b+=7*(8>$b=ord(max(str_split($s)))-54);echo base_convert($s,$b,10),",$b";}
but a return value is easier to test:
function test($x,$y,$e){static $h='<table border=1><tr><th>input</th><th>output</th><th>expected</th><th>ok?</th></tr>';echo"$h<tr><td>$x</td><td>$y</td><td>$e</td><td>",(strcmp($y,$e)?'N':'Y'),"</td></tr>";$h='';}
foreach([1=>'1,2','HELLOWORLD'=>'809608041709942,33','ILOVECODEGOLF'=>'21537319965321352644,32','POTATO'=>'627732894,30','1FISH2FISH'=>'22338345482731,29']as$x=>$e)test($x,implode(',',f($x)),$e);echo'</table>';