You can do it simply through css. On header you have div named site-logo. Inside that div put two logos with different class names. Now 1 logo will be display none all the times except on your particular page. On body class wp automatically generated page id or cat id, in your case body.category-8 .(your normal logo classname) should be display none and other image should be display block.
this is simple one, you can do it through jquery also.