Multiple logo’s in wordpress menu

You can use an if statement in the header

if ( is_home() ) {
    <img src="https://wordpress.stackexchange.com/home-image.png"/>
} else {
   <img src="/other-image.png"/>
}

Below are a few more way to pick a specific page

// When any single Page is being displayed.
is_page();

// When Page 42 (ID) is being displayed.
is_page( 42 );

// When the Page with a post_title of "Contact" is being displayed.
is_page( 'Contact' );

// When the Page with a post_name (slug) of "about-me" is being displayed.
is_page( 'about-me' );