Mark’s idea can work, but logos usually are output from the header template, the same file is used for all pages. Having unique header files for each template gets cumbersome and maintaining the header content gets more complicated when many different files are involved.
JS can work too, but you typically see the old logo briefly before the proper one gets swapped in. Not the best user experience.
I would dynamically output the proper logo in PHP based on the requested page. The specifics depend on how your theme outputs a logo. In some cases it is a pluggable function. You can redefine the same function is a child theme and it’ll be used instead of the original. Before doing that, see if there is a filter you can hook to alter the image src. Filter hooks are preferable to redefining pluggable functions. In PHP you cannot redefine functions unless the theme dev purposely made the function pluggable. Pluggable function declarations are wrapped in code that’s something like this:
if ( ! function_exists( 'foobar_theme_logo' ) ):
//foobar_theme_logo function declaration here
endif;
The way code can determine the correct image src varies. You can get the requested object ID with get_queried_object_id(). With the ID, a switch/case structure can determine the correct image src. Or the data can be stored in an array keyed by ID. Get the associated value out of the array. The image src could be saved as post meta, simply get the meta value with the ID. If there is no match or the returned value is empty, just output the default logo.
