Why is everyone trying to do it the hard way? Sure, some of these approaches will work, but they're more complicated than is necessary.
OK, first - you need an image that will fit within your navbar. You can adjust your image via css height attribute (allowing width to scale) or you can just use an appropriately sized image. Whatever you decide to do - the way this looks will depend on how well you size your image.
For some reason, everyone wants to stick the image inside of an anchor with navbar-brand, and this isn't necessary. navbar-brand applies text styles that aren't necessary to an image, as well as the navbar-left class (just like pull-left, but for use in a navbar). All you really need is to add navbar-left.
<a href="#" class="navbar-left"><img src="/path/to/image.png"></a>
You can even follow this with a navbar-brand item, which will appear to the right of the image.
Answer from Michael on Stack OverflowVideos
Why is everyone trying to do it the hard way? Sure, some of these approaches will work, but they're more complicated than is necessary.
OK, first - you need an image that will fit within your navbar. You can adjust your image via css height attribute (allowing width to scale) or you can just use an appropriately sized image. Whatever you decide to do - the way this looks will depend on how well you size your image.
For some reason, everyone wants to stick the image inside of an anchor with navbar-brand, and this isn't necessary. navbar-brand applies text styles that aren't necessary to an image, as well as the navbar-left class (just like pull-left, but for use in a navbar). All you really need is to add navbar-left.
<a href="#" class="navbar-left"><img src="/path/to/image.png"></a>
You can even follow this with a navbar-brand item, which will appear to the right of the image.
<div class="navbar-header">
<button type="button" class="navbar-toggle" data-toggle="collapse"
data-target=".navbar-ex1-collapse">
<span class="sr-only">Toggle navigation</span>
<span class="icon-bar"></span>
<span class="icon-bar"></span>
<span class="icon-bar"></span>
</button>
<a class="navbar-brand" rel="home" href="#" title="Buy Sell Rent Everyting">
<img style="max-width:100px; margin-top: -7px;"
src="/img/transparent-white-logo.png">
</a>
</div>
To keep your header image inside your header at all times you can simply apply max-height: 100%; to your image. This will stop it from overflowing its container. Here is an example: https://jsfiddle.net/rktr4q5v/1/
You can see that the images are both quite small. This is because the class navbar-brand has a height of 50px applied. If you remove this then your images will expand to their original width but remain inside the navbar. Here is an example of this: https://jsfiddle.net/rktr4q5v/2/
If you need to do specific size styling for your logos then you will need to use media queries to change the size of your logo at different screen widths. The documentation for this can be found here and a beginners guide can be found here
I have also fixed some of your mistakes. You can refer this.
Adjust your Logo Size in navbar-brand style tag. If requires
<!DOCTYPE html>
<html lang="en">
<head>
<title>Test</title>
<link rel="stylesheet" href="https://maxcdn.bootstrapcdn.com/bootstrap/3.3.7/css/bootstrap.min.css">
<link rel="stylesheet" type="text/css" href="test2.css">
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1.0">
</head>
<body>
<div class="navbar navbar-default " role="navigation">
<div class="container-fluid">
<div class="navbar-header">
<button type="button" class="navbar-toggle" data-toggle="collapse" data-target=".navbar-collapse">
<span class="sr-only">Toggle Navigation</span>
<span class="icon-bar"></span>
<span class="icon-bar"></span>
<span class="icon-bar"></span>
</button>
<a class="navbar-brand" href="#"><img src="https://static.pexels.com/photos/34950/pexels-photo.jpg" class="img-responsive img-rounded" style="max-height: 40px; margin-top: -15%;"></a>
</div>
<div class="navbar-collapse collapse navbar-right">
<ul class="nav navbar-nav">
<li><a href="#">HOME</a></li>
<li><a href="#">LINK</a></li>
<li><a href="#">LINK</a></li>
<li><a href="#">LINK</a></li>
<li><a href="#">LINK</a></li>
</ul>
</div>
</div>
</div>
<script src="https://ajax.googleapis.com/ajax/libs/jquery/3.3.1/jquery.min.js"></script>
<script src="https://maxcdn.bootstrapcdn.com/bootstrap/3.3.7/js/bootstrap.min.js"></script>
</body>
</html>