If I read your question well, you already use WordPress Custom Logo, but you want a different logo on your front page. This means you will need to upload a separate image.
Assuming you already know how to use the customizer in general you will need to build an upload field like this:
$wp_customize->add_control(
new WP_Customize_Upload_Control(
$wp_customize,
'wpse238690_front_logo',
array(
'label' => __( 'First image', 'wpse215632_theme' ),
'description' => __( 'More about first image', 'wpse215632_theme' ),
'section' => 'wpse238690__section_id',
'settings' => 'wpse238690__setting_id',
) )
);
Replace the section and setting to fit the control in the right place in your customizer system.
In your header template you must then include a conditional with is_front_page: display the normal logo in the header if you’re not on the front page. In your front page template include the logo with get_mod('wpse238690_front_logo').