Our homepage slider has a different colour than the rest of our site so the black logo we have that works on all our other pages is invisible on our home page, is there a way to insert code that allows us to swap out our logo for a white one (or just straight up change the colour to white) specifically for our home page?
Using Brooklyn theme
Such things usually occur as a 'template' that the rest of your site gets served up in, allowing easy uniformity across the space. Thusly, changing the header on one will change the header on all.
A reasonably easy workaround would be to have a standalone landing page, clone the existing homepage make the required edits so it features the logo you desire and change all the other appropriate links sitewide to point to that instead.