That's because you're trying to call the method log_file on an undefined value in $exp. Check for errors where you create $exp. Also keep in mind that things like PATH and other environment variables can have quite different values than your normal login shell provides.

#!/bin/sh

logname=$USER
home=$HOME

for i in `env | sed 's/=.*//'`; do
    if [ X"$i" != XPATH ]; then
        unset $i
    fi
done

export LOGNAME=$logname
export TERM=dumb
export HOSTTYPE=i386
export PATH=/usr/bin:/bin
export HOME=$home
export SHELL=/bin/sh
export OSTYPE=Linux
export SHLVL=1
export _=/usr/bin/env

exec bash -noprofile -norc $*
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Going the other way, you can use RE (tilly) 3: Get default login environment in your cron jobs to make their environments match what you normally see when you are logged in.

Incidentally your script above is easy to write in Perl:

#! /usr/bin/perl

%ENV = (
  LOGNAME   => $ENV{LOGNAME},
  TERM      => "dumb",
  HOSTTYPE  => "i386",
  PATH      => "/usr/bin:/bin",
  HOME      => $ENV{HOME},
  SHELL     => "/bin/sh",
  OSTYPE    => "Linux",
  SHLVL     => 1,
  _         => "/usr/bin/env",
);

exec("bash", "-noprofile", "-norc", @ARGV);
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UPDATE: As pointed out by AM below, I was setting HOME twice. It makes more sense not to, so I've fixed that.

Why is HOME set twice? Is it (so) important?