# Simple Equations with the Natural Base

^{x}where the base b > 1 and x is any real number.

In many situations the base e is used. The base e is called the natural base and is an irrational number that is approximately 2.718281828.

The natural exponential function has the form:

NATURAL EXPONENTIAL FUNCTION

y = ae^{x}

Where a ≠ 0.

Some examples are:

1. y = e

^{x}(Where a = 1)

2. y = 65e

^{x}(Where a = 65)

3. y = -3e

^{x}(Where a = -3)

The properties for the natural base are:

**Property 1:**e

^{0}= 1

**Property 2:**e

^{1}= e

**Property 3:**e

^{x}= e

^{y}if and only if x = y

**One-to One Property**

**Property 4:**ln e

^{x}= x

**Inverse Property**

Just as logarithms are inverse functions to exponents, the inverse function to e

^{x}is ln x, called the

**. This is shown in Property 4.**

__natural log__Let's solve some simple natural exponential equations:

^{x}= e

^{12}

Properties 1 and 2 do not apply, as the exponent is neither 0 nor 1. Since both terms are natural exponents, Property 3 is the most appropriate. |
Property 3 - One to One |

The equation is already written in the form of b |
e |

Property 3 states e |
x = 12 |

**Example 2: e**

^{x}= 41
Properties 1 and 2 do not apply, as the exponent is neither 0 nor 1. Since 41 can't accurately be written as an exponent with base e, the most appropriate property is the Inverse property, Property 4 |
Property 4 - Inverse |

To apply Property 4, take the |
ln e |

Property 4 states that ln e |
x = ln 41 |

Related Links:Math algebra Complex Equations with the Natural Base Introduction and Simple Equations with the Natural Base Algebra Topics |

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