It can be done with CSS, but the logo won’t be clickable like your site title; is that OK? Just add this to the end of your custom CSS:
#site-title, #site-description {
font-size: 0;
}
#site-title:before {
content:url(https://www.temasekclub.com.au/wp-content/uploads/2013/08/box-singapore.jpg);
}
Change the URL value to point to whatever image file you’d like. The first CSS rule hides the title & tag line by setting the font size to zero, then the second rule inserts the image before the site title.
By the way, you have a syntax error in your custom CSS. I see this at the end of your custom CSS:
.three-column #secondary {
width: 44.89%;
.widget-area,
.widget-area input {
font-size: 15px;
line-height: 18px;
You need to add the closing brace } for each of those rules, or those CSS rules, and any CSS which follows, will not be read.
If, however, you would like your logo to be clickable, i.e., it takes your user back to your home page (the logo for most sites do this), you can easily replace the site title with a clickable logo by inserting a single line of JQuery code. Since you don’t want to create a child theme, you can add this code using a scripting plugin, like Header and Footer:
<script>
jQuery(document).ready(function($){
$("#site-title span a").html('<img src="https://www.temasekclub.com.au/wp-content/uploads/2013/08/box-singapore.jpg" />');
});
</script>
This will replace the contents of the site title with an image (again, replace the src value with the file you want to point to). You will also have to hide the site description with this CSS:
#site-description {
display: none;
}