These seem scary at first, but they’re not actually too bad.
A logarithm has a "base", and a "power". When is written, it means "what needs to be raised to the power of to get ?" For example, , as .
The definition of a logarithm is that if and only if . From here, we can prove a bunch of facts about the logarithm function.
For example, if we let and then we can then express in terms of and .
We can then use one of the law of powers, that 22 2 This is explored above. to write that
After this, we can use the definition of the function to simplify the right-hand side of the previous equation.
And from our earlier definitions of and we can say that 33 3 This is particularly powerful because it means that we can write any multiplication as a sum (and there’s a lot more algebra that can be applied to sums than products).