What is it about the number e that makes it so that taking log to the base e is called the natural logarithm?
In: Mathematics
First: A logarithm must have a base associated with it. Let’s say that base is 10. Then taking the base 10 logarithm of a number is the answer to the question, To what power should I raise 10 to get this number? So if you are taking the base 10 logarithm of 100, the answer is 2, because 10 raised to the power of 2 equals 100.
The number e shows up in a lot of different contexts, but one of the simplest ones is that the function e^x has a rate of change that is its value. This makes it a simple to use base for describing all kinds of things where the amount they are changing is based on their value, such as population growth, growth due to interest rates, air resistance and more.
So the function e^x shows up so much we give it it’s own name, the exponential function. And because it shows up so much we are often trying to solve for x where e^x = some number. So we need to take the logarithm of both sides of our equation to answer this, and the most natural base to use for this is the number that is being raised to the power of x, which is e. So we use a logarithm with a base of e and we call it the natural logarithm. It also gets called “log base e”, but since it shows up so often people just call it the natural log.
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