I can propose you this (for the logo, I add one and I give a size, but check this proposition with your own logo):
img{
height: 50px; /*ADD*/
}
.navbar {
background-color: #e7e7e7;
display: flex; /*ADD*/
align-items: center; /*ADD*/
}
.navbar-element {
/*display: inline-block;*/
color: #202020;
padding: 10px;
text-decoration: none;
}
.navbar-element:hover {
color: #000000;
background-color: #d0d0d0;
}
.navbar-element.active {
color: #f0f0f0;
background-color: #006000;
}
.navbar-element.active:hover {
color: #ffffff;
background-color: #007000;
}
<nav class="navbar">
<a class="navbar-element" href="."><img class="logo-navbar" src="http://www.bardfieldacademy.org/wp-content/uploads/2016/08/00106-3D-company-logos-design-free-logo-online-02.png"></a>
<a class="navbar-element active" href=".">Startseite</a>
<a class="navbar-element" href="./about-us">über uns</a>
<a class="navbar-element" href="./shop">Shop</a>
</nav>
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