# Complex Equations with the Natural Base

**EXPONENTIAL EQUATIONS: Simple Equations With the Natural Base**.

This discussion will focus on solving more complex problems involving the natural base. Below is a quick review of natural exponential functions.

**Quick Review**

The natural exponential function has the form:

NATURAL EXPONENTIAL FUNCTION

y = ae^{x}

Where a ≠ 0

The natural base e is an irrational number, like π, which has an approximate value of 2.718.

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**

Let's solve some complex natural exponential equations.

Remember when solving for x, regardless of the function type, the goal is to isolate the x-variable.

^{x}-12 = 47

In this case add 12 to both sides of the equation. |
e |

Since the x is an exponent of natural base e, take the natural log of both sides of the equation to isolate the x-variable, Property 4 - Inverse. |
ln e |

Property 4 states |
x = ln 59 Apply Property x = ln 59 Exact answer $x\approx 4.078$ Approximation |

**Example 1: 3e**

^{2x-5}+ 11 = 56
In this case subtract 11 from both sides of the equation. Then divide both sides by 3. |
3e
3e
e |

Since the x is an exponent of natural base e, take the natural log of both sides of the equation to isolate the x-variable, Property 4 - Inverse. |
ln e |

Property 4 states that ln e Next isolate the x but adding 5 and dividing by 2. |
2x - 5 = ln 15 Apply Property 2x = ln 15 + 5 Add 5 $x=\frac{\mathrm{ln}15+5}{2}$ Divide by 2 $x=\frac{\mathrm{ln}15+5}{2}$ Exact answer $x\approx 3.854$ Approximation |

**Example 2: 1500e**

^{-7x}= 300
In this case divide both sides of the equation by 1500 |
1500e
e |

Since the x is an exponent of natural base e, take the natural log of both sides of the equation to isolate the x-variable, Property 4 - Inverse. |
ln e |

Property 4 states that ln e Thus the left-hand side simplifies to the exponent, -7x. Next isolate the x but dividing by -7. |
-7x = ln 0.2 Apply Property $x=\frac{\mathrm{ln}0.2}{-7}$ Divide by -7 $x=\frac{\mathrm{ln}0.2}{-7}$ Exact answer $x\approx 0.230$ Approximation |

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

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