What is a natural logarithm? Why is ln(1) = 0?
In: Mathematics
log(1) = 0 is true of all logs, because log is the inverse of exponent, and any number to the power of zero is 1.
It can be a little confusing, but “the log in base X of Y equals Z” is the same as saying “X to the power of Z equals Y.” You may have to read that several times and maybe even write it out for it to sink in.
The natural log, usually written ln (but to many programmers and mathematicians “log” by default means ln), is the log using base e. “e” is a special number, sort of like pi, that is special because of how it relates to rates of change. It’s about equal to 2.718 plus infinite digits after that.
The function e^x is special because if you graph it, the rate of change at every point is also equal to e^x. The function describes its own rate of change.
This is useful when modeling situations where the rate of change depends on how much of something you currently have.
For example, gaining interest on money. If you have more money, you earn more interest so your amount of money increases faster, and then you have more money and get even more interest and get even more money…
How fast the money grows depends on how much money there currently is. The rate of change depends on the amount, and the number e is special because when you use it as a base, the rate of change is *exactly equal* to the current amount.
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