# Using the Natural Base e: Definition & Overview

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• 0:02 The Natural Base e
• 1:14 The Natural Logarithm
• 2:29 The Graph
• 2:49 Uses
• 3:40 Lesson Summary

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Lesson Transcript
Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

In this video lesson, you will learn what kind of number 'e' is. Also learn the formal name for this unique number. See what kind of graph the natural base 'e' graphs into, as well as the uses of these types of functions.

## The Natural Base e

e is an interesting number. e is an irrational number that begins with 2.718281828459045 and continues indefinitely with no pattern. An irrational number is a number that cannot be written as a fraction and therefore has numbers that never repeat. It can be fun and impressive to memorize the beginning part of e. Some use a memory aid such as coming up with a phrase where each word matches each number in the number of letters it has.

For example, the phrase 'to express e, remember to memorize a sentence to simplify this' matches each word to each number. The beginning word has two letters matching it with the first number, 2. We place a dot after the two. Then the next word 'express' has seven letters matching it with the second number, 7. The following words continue in the same manner. If you want to memorize it and impress your friends, you can use this memory aid or any other memory aid that you know of.

## The Natural Logarithm

When the number e is used as the base for a logarithm, we call the number e the natural base, and the logarithm is called a natural logarithm. Usually, logarithms are assumed to have a base of 10, so when we change the base to e, we write it as ln(x) or we write log with a subscript of e to show that it has e as the base instead of 10.

The way the natural log works is if we take the natural log of x, it will be equal to the power of e that gives you x. For example, the natural log of e is 1 since e to the first power is e. The natural log of 1, on the other hand, equals 0 because e has to be raised to the 0th power for it to equal 1.

The inverse function of the natural log of x is simply e^x. The inverse is the function that reverses the original function. So, this means that if ln(x) gives you y, then e^y will give you x. They reverse each other.

## The Graph

When graphed, the natural log of x curves and slowly approaches infinity as x increases. The function never touches the y-axis but goes to negative infinity the closer the function gets to x = 0.

The y-axis is therefore an asymptote of the natural log function.

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