HeaderMenu.vue

<template>
    <v-btn v-if="!isChangingRoute" icon key="headerMenu">
        <i class="fas fa-bars" key="toolbarBars" v-if="isHomeRoute"></i>
        <i class="fas fa-arrow-left" key="toolbarArrow" v-else></i>
    </v-btn>
</template>
<script>
  export default {
    name: 'app-header-menu',
    created(){
      this.isHomeRoute()     
    },
    data () {
      return {
        isChangingRoute: true
      }
    },
    methods:{
      isHomeRoute: function () {
       if(this.$route.name === 'home'){
         return this.isChangingRoute = true
       }
       return this.isChangingRoute = false
      }
    },
    watch:{
        '$route'(to, from){
            this.isChangingRoute = this.isHomeRoute()
        }
    }
  }
</script>