https://youtu.be/vGsOA2eqkig

In this video I go over the natural exponential function y = e<sup>x</sup> and define it strictly from looking at the inverse of the natural logarithmic function y = ln(x). The natural exponential function is one of the most widely used and important function in all of mathematics so it is important to understand the definition of it.

---

Watch Video On:

- DTube: [https://d.tube/#!/v/mes/wz4b4tjsk3b](https://d.tube/#!/v/mes/wz4b4tjsk3b)
- BitChute: [https://www.bitchute.com/video/CcPnEoatbeSu/](https://www.bitchute.com/video/CcPnEoatbeSu/)
- YouTube: [https://youtu.be/vGsOA2eqkig](https://youtu.be/vGsOA2eqkig)

Download Video Notes: https://1drv.ms/b/s!As32ynv0LoaIg4QqF_gaw8_b_k3vCw?e=2s2TqK

---

# View Video Notes Below!

---

>Download these notes: Link is in video description.
>View these notes as an article: https://peakd.com/@mes
>Subscribe via email: http://mes.fm/subscribe
>Donate! :) https://mes.fm/donate 
>
>Reuse of my videos:
>- Feel free to make use of / re-upload / monetize my videos as long as you provide a link to the original video.
>
>Fight back against censorship:
>- Bookmark sites/channels/accounts and check periodically
>- Remember to always archive website pages in case they get deleted/changed.
>
>Join my private Discord chat room: https://mes.fm/chatroom
>
>Check out my Reddit and Voat math forums:
>- Reddit: https://reddit.com/r/AMAZINGMathStuff
>- Voat: https://voat.co/v/AMAZINGMathStuff
>
>Buy "Where Did The Towers Go?" by Dr. Judy Wood: https://mes.fm/judywoodbook
>Follow along my epic video series:
>- #MESScience: https://mes.fm/science-playlist
>- #MESExperiments: https://peakd.com/mesexperiments/@mes/list
>- #AntiGravity: https://peakd.com/antigravity/@mes/series
>-- See Part 6 for my Self Appointed PhD and #MESDuality breakthrough concept!
>- #FreeEnergy: https://mes.fm/freeenergy-playlist
>
>---
>
>NOTE #1: If you don't have time to watch this whole video:
>
>- Skip to the end for Summary and Conclusions (if available)
>- Play this video at a faster speed.
>-- TOP SECRET LIFE HACK: Your brain gets used to faster speed. (#Try2xSpeed)
>-- Try 4X+ speed by browser extensions or modifying source code.
>-- **Browser extension recommendation: https://mes.fm/videospeed-extension**
>-- See my tutorial to learn more: https://peakd.com/video/@mes/play-videos-at-faster-or-slower-speeds-on-any-website
>- Download and read video notes.
>- Read notes on the Hive blockchain #Hive
>- Watch the video in parts.
>
>NOTE #2: If video volume is too low at any part of the video:
>
>- Download this browser extension recommendation: https://mes.fm/volume-extension

---

# The Natural Exponential Function y = e<sup>x</sup>

![Natural Exponential ex.jpeg](https://files.peakd.com/file/peakd-hive/mes/CteYxZNh-Natural20Exponential20e5Ex.jpeg)

**The natural exponential function y = e<sup>x</sup> derived from the natural logarithm function y = ln(x)**

![image.png](https://files.peakd.com/file/peakd-hive/mes/EwtbnXSU-image.png)
	
ln(x) is an increasing function thus it is a one-to-one function (i.e. each y-value has only 1 x-value)

Thus it has an inverse function (i.e. can switch or “invert” all data points from (x, y) to (y, x).

Let exp be the inverse of y = ln(x), in other words:

![image.png](https://files.peakd.com/file/peakd-hive/mes/Uc4N32B4-image.png)

Recall the cancellation formulas:

![image.png](https://files.peakd.com/file/peakd-hive/mes/Sp5fnGYS-image.png)

![image.png](https://files.peakd.com/file/peakd-hive/mes/zdIh1WZI-image.png)

Recall also the log law: If r is a rational number:

![image.png](https://files.peakd.com/file/peakd-hive/mes/Hlr0yVDU-image.png)

Taking into account all the above requirements for the inverse of ln(x), it leads us to define (even if x is irrational).

![image.png](https://files.peakd.com/file/peakd-hive/mes/mKX0KGWI-image.png)

The natural exponential function, ex, is very important because it is used throughout calculus, engineering, etc. 
![image.png](https://files.peakd.com/file/peakd-hive/mes/D8G1TsuU-image.png)