You need to upload your svg to the assets folder. In my example i call the file logo.svg

Then edit the header.liquid file

In the store files edit header.liquid and locate the code between
{% if section.settings.logo %}

{place the new code here}

{% else %}
other code
{% endif %}

Then place this following code between it

<a href="{{ routes.root_url }}" class="site-header__logo-image{% if section.settings.align_logo == 'center' %} site-header__logo-image--centered{% endif %}">
  {% capture image_size %}{{ section.settings.logo_max_width | escape }}{% endcapture %}
  <img src="{{ 'logo.svg' | asset_url }}" 
  alt="{{ shop.name }}"
  style="max-width: {{ image_size }}px;">
</a>