### A Logarithm is Just the Number of Digits

If you don’t have a good intuition for how logarithms work, here is how you can explain it to a third grader: It’s the number of digits (roughly). For example it’s the number of zeros in these:

For this to work out we have to transition through zero:

If it bothers you that I’m off by one, because clearly 100 has 3 digits, not 2, you can get the actual number of digits by rounding away from 0 to the next integer.

Where it gets complicated is when you have a number between two round numbers:

We can’t just count the number of digits. It’s got to be somewhere between the other two numbers, coming out to roughly . But lets not worry about those odd cases for now. Where this really helps is when trying to build an intuition for the logarithm rules:

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