How to update a component based on the theme?

[mikwee]

I'm trying to change the logo based on the theme. For that, I wrote this component:

<script lang="ts">
  import { onMount } from 'svelte'
  import { tick } from 'svelte/internal'
  import { theme } from '../store/theme'

  async function printTheme() {
    await tick();
    console.log("Logo theme: " + $theme);
  }

  let path;
  theme.subscribe(theme => {
    switch (theme) {
      case 'dark':
        console.log("Dark theme triggered");
        path = '/assets/Ein Sof Dark.svg'; // White logo
      case 'light':
        console.log("Light theme triggered");
        path = '/assets/Ein Sof.svg'; // Black logo
    }
    printTheme();
  });

  onMount(() => {
    printTheme();
  });
</script>

<img class="header__logo-img" src={path} alt="The Ein Sof logo"/>

For some reason, the logo stays black, even when the theme is dark. This is the terminal output:

14:55:17 [200] / 27ms
Dark theme triggered
Light theme triggered
Logo theme: dark

Can you perhaps help me understand how the theme works? Thanks.