Logarithms vs Radicals vs Exponents (The True Difference)
As a student, whenever I would do PEMDAS operations I felt solid. I understood inverses (division undoes multiplication and vice versa) and how to check my answers be reversing my math (5 - 3 = 2 so 2 + 3 must equal 5). I even felt comfortable with exponents and treating radicals as their inverse and back (vice versa). But when Logarithms entered the picture, I felt the ground drop out underneath me. I couldn’t make the connections anymore and I was defaulting to memorization — something I hated doing as someone who considers active learning and connections way more valuable and purposeful than passive memorization (even as a student, before I could articulate this feeling). But I left it alone because it seemed too huge to untangle on my own.
If you find yourself in the same boat and feel like Logarithms humble you every time you work with them, then this guide is for you. We are going to uncover that mystery and pick them apart until there isn’t anything confusing left.
What question do Logarithm’s answer?
Math is really built up from asking questions, then asking more complex questions on top of questions. Sorry for that terrible sentence, but let me explain first. Think about the progression from addition to multiplication to exponents:
Addition asks:
If I have 3 and add 4, what do I get?
3 + 4 = ?
Answer: 7
Subtraction asks the inverse question:
What number do I need to add to 3 to get 7?
3 + ? = 7
Answer: 4 (because 7 - 3 = 4)