Custom Master page dynamic logo using Visual studio SharePoint 2013

Question

We have used a starter master page and created a visual studio master page solution. The master page has OOB global level navigation.

These navigation have sites(Site A, Site B, SIte C) in them and in-turn have subsite(A1,A2,A3;B1,B2,B3;C1,C2,C3). On the site collection page we have a logo1. But when the user clicks on site lets say Site B, the logo should change in the master page to LogoB. Likewise when the user clicks on Site C

How can this be achieve?

Here is the code for the Link Button

<div class="ms-dialogHidden">
<SharePoint:AjaxDelta ID="Custom_Logo" BlockElement="true" runat="server"><SharePoint:SPLinkButton runat="server" NavigateUrl="~sitecollection/">
<SharePoint:SiteLogoImage ID="xxxxxxxxxxxxxxxxxxxx" LogoImageUrl="&lt;% $SPUrl:~sitecollection/_catalogs/masterpage/StarterBranding/logo.jpg %&gt;" AlternateText="Back to Home" runat="server" />
</SharePoint:SPLinkButton>
</SharePoint:AjaxDelta> </div>

Answer 1

OK, I completely edited my answer, see answer versioning to get the comlete previous one.

Now that I've seen your markup, here what I think:
You should use (note ~site instead of ~sitecollection):
<SharePoint:SiteLogoImage ID="xxxxxxxxxxxxxxxxxxxx" LogoImageUrl="&lt;% $SPUrl:~site/_catalogs/masterpage/StarterBranding/logo.jpg %&gt;" AlternateText="Back to Home" runat="server" />

And you manually upload a different logo.jpg in each sub-site, at in _catalogs/masterpage/StarterBranding.
This way, even if you create new sites later, it will be easy to upload the corresponding image at this location.