Can I specify random logos?

You could do it with JS i suppose, here is code i hope it works

<script>
if (document.getElementById) { window.onload = swap };
function swap() {
var numimages=4;
rndimg = new Array("IMAGE URL 1", "IMAGE URL 2", "IMAGE URL 3", "IMAGE URL 4");
x=(Math.floor(Math.random()*numimages));
randomimage=(rndimg[x]);
document.getElementById("banner").style.logo-big = "url("+ randomimage +")";
}
</script>

PS this only works on hard refresh. also the css to big logo would be

.logo-big { 
background-image: url('DEFAULT LOGO IMAGE URL HERE');
}

The JS would work because i am using that to rotate my banner image randomly on every hard refresh not sure the exact code for logo CSS.