Let your theme's customizer logo upload option override the default WordPress login logo

functions-custom-login-logo.php

<?php
/*
 * Change the "mnml" text througout the code below to whatever you'd like to use
 *
 * Also, make sure the get_theme_mod instances are changed to your customizer option names
 *
 */

// Add custom logo to login page if it's uploaded through the customizer
if ( get_theme_mod( 'mnml_logo' ) )  {
	
function mnml_login_logo() {
	
	list($width,$height) = getimagesize(get_theme_mod( "mnml_logo" ));
	
	echo '<style type="text/css">
		#login {
			width: '. $width .'px !important;
		}
        	h1 a {
			background-image:url('. get_theme_mod( "mnml_logo" ) .') !important;
			background-size: '. $width .'px !important;
			width: '. $width .'px !important;
			height: '. $height .'px !important;
		}
	</style>';
}

add_action('login_head', 'mnml_login_logo');

// Change login page logo link to the website home page
function mnml_login_logo_url() {
    return get_bloginfo( 'url' );
}
add_filter( 'login_headerurl', 'mnml_login_logo_url' );

// Change login page logo title to the website name
function mnml_login_logo_url_title() {
    return get_bloginfo( 'name' );
}
add_filter( 'login_headertitle', 'mnml_login_logo_url_title' );

} // if get_theme_mod('mnml_logo')
?>