What is a logarithm?
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A base raised to an exponent gives you a number. But suppose you only have the base and the result. You would use a logarithm to figure out what the exponent had been. That’s where it comes from. In terms of why it’s useful, it’s not so much about solving for the exponent of particular equations as much as it is about modeling relationships between numbers that experience exponential growth or decay. So you would use log(x) a lot in describing those things, kind of like how you would use doubling a lot, or 2(x), to describe a process like mitosis.
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