This is not quite as straightforward as it ought to be (given that you want to keep the Log In button, which is rendered by the same function). Put the following in your plugins/bp-custom.php file:
function bbg_customize_adminbar_hooks() {
remove_action( 'bp_adminbar_menus', 'bp_adminbar_login_menu', 2 );
}
add_action( 'bp_loaded', 'bbg_customize_adminbar_hooks', 999 );
function bbg_adminbar_login_menu() {
global $bp;
if ( is_user_logged_in() )
return false;
echo '<li class="bp-login no-arrow"><a>root_domain . '/wp-login.php?redirect_to=' . urlencode( $bp->root_domain ) . '">' . __( 'Log In', 'buddypress' ) . '</a></li>';
}
add_action( 'bp_adminbar_menus', 'bbg_adminbar_login_menu', 2 );
Here’s a bit more about bp-custom.php, in case you haven’t used it before https://codex.buddypress.org/extending-buddypress/bp-custom-php/
if you simply want to hide Admin bar login and sign up, just add
remove_action( 'bp_adminbar_menus', 'bp_adminbar_login_menu', 2 );
to your functions.php and login and sign up will go away
