The simplest way is to use conditional tags is_page(),
For example
<div class="row-fluid top-banner">
<div class="container">
<div class="banner-overlay"></div>
<?php
$logo = of_get_option('logo', '' );
if ( !empty( $logo ) ) { ?>
if ( is_page( 'contact' ) ) {
<a class="brand brand-image" href="https://wordpress.stackexchange.com/questions/114494/<?php echo esc_url( home_url("/contact/' ) ); ?>" title="<?php echo esc_attr( get_bloginfo( 'name', 'display' ) ); ?>" rel="contact"><img src="<?php echo $newsource; ?>" alt="<?php echo esc_attr( get_bloginfo( 'name', 'display' ) ); ?>"><h1><?php if(!of_get_option('disable_description')){ ?><small><?php bloginfo( 'description' ); ?></small><?php } ?></h1></a>
} else {
<a class="brand brand-image" href="https://wordpress.stackexchange.com/questions/114494/<?php echo esc_url( home_url("https://wordpress.stackexchange.com/" ) ); ?>" title="<?php echo esc_attr( get_bloginfo( 'name', 'display' ) ); ?>" rel="home"><img src="<?php echo $logo; ?>" alt="<?php echo esc_attr( get_bloginfo( 'name', 'display' ) ); ?>"><h1><?php if(!of_get_option('disable_description')){ ?><small><?php bloginfo( 'description' ); ?></small><?php } ?></h1></a>
}
<?php }else{ ?>
<a class="brand brand-text" href="<?php echo home_url( "https://wordpress.stackexchange.com/" ); ?>" title="<?php echo esc_attr( get_bloginfo( 'name', 'display' ) ); ?>" rel="home"><h1><?php bloginfo( 'name' ); ?><?php if(!of_get_option('disable_description')){ ?><small><?php bloginfo( 'description' ); ?></small><?php } ?></h1></a>
<?php }
if(of_get_option('disable_description')){ $top_banner_fix = 'style="top:15px;"'; }else{ $top_banner_fix = ''; }
?>
</div>
In the conditional tag is_page( ‘contact’ ) you can see i have added a new image source which would show a new image in the contact page.
You can also use ids, slugs of the page and also can pass an array.
http://codex.wordpress.org/Function_Reference/is_page
Hope it helps!