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The Magic Initiate feat gives access to two cantrips from a class. It says

Choose a class: bard, cleric, druid, sorcerer, warlock, or wizard. You learn two cantrips of your choice from that class's spell list.

It also says

Your spellcasting ability for these spells depends on the class you chose: Charisma for bard, sorcerer, or warlock: Wisdom for cleric or druid; or Intelligence for Wizard.

I assume that spell attacks worked the same way - i.e. so if I chose Sorcerer and had a +2 charisma modifier and a +3 Proficiency Bonus, then if I Fire Bolt someone then the spell attack bonus would be +5 versus AC.

However, cantrips scale up with your level. Fire Bolt says

This spell's damage increases by 1d10 when you reach 5th level (2d10), 11th level (3d10), and 17th level (4d10).

So if I'm a Level 5 Rogue (not Arcane Trickster), then does a Fire Bolt I cast deal 2d10 damage on a hit?

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Your understanding is correct

The Magic Initiate feat, as you've correctly noted, lets you learn two Cantrips, and Cantrips scale with your character level, not your caster level. This means that your Initiate-Rogue can get the full, scaling power of their Cantrips for the rest of their life! You are also correct on how to calculate your spell attack bonus, using the relevant Ability modifier and your proficiency bonus.

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The wording in each cantrip spell indicates that it applies your character level, not your class level. Thus, cantrips increase based on character level, and your assumption is correct -- when your character level reaches 5, your Fire bolt damage will increase to 2d10.

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Note in the text you quoted that the level is not qualified.

This spell's damage increases by 1d10 when you reach 5th level (2d10), 11th level (3d10), and 17th level (4d10).

If you go through the other cantrips you will see that they are worded similarly. So RAW means the level you cast the cantrip is at is based on your character's overall level.

However if you read the rules on multi-classing on page 163 of the PHB. It says

If you multiclass but have the Spellcasting feature from only one class, you follow the rules as described in that class

However this is qualified by the statement If you multiclass. That is not the case here. Here you are taking a feat. So my ruling would be that you cast the cantrip at your character's overall level.

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