https://youtu.be/c4R2urO1GCU

In this video I go over general exponential functions and look at exponential functions with base a, such as a^x and derive some exponential and logarithmic laws that can be used as the basis for exponent laws that I will prove in later videos. So stay tuned!

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# General Exponential Functions

![General Exponential Functions.jpeg](https://cdn.steemitimages.com/DQmZfdnPu3PbTB2hpxVrGxYfyHjntQ6d2Y8pD2dUXyGpcfZ/General%20Exponential%20Functions.jpeg)

If a > 0 and r is any rational number then by the laws I covered in my earlier videos: 

![image.png](https://cdn.steemitimages.com/DQmfLax38Wi43t7umu2pT8oU5YBR6VW6bmq5Rd6M5wTwPJN/image.png)

Because we combined two laws, one being for all real numbers greater than 0 and the other being for rational numbers, we can therefore define for all real numbers (even for irrational numbers):

![image.png](https://cdn.steemitimages.com/DQmdYELshnaDd4QoG7cyN2atnPmrCuj4hpd4iM9YWWEE9zt/image.png)

The function f(x) = a<sup>x</sup> is called the **exponential function with base a.**

Notice that a<sup>x</sup> is positive because e<sup>x</sup> is positive.

This definition allows to us to extend another law that I already showed in my earlier video:

![image.png](https://cdn.steemitimages.com/DQmWteQ8i7Ru78yqf2Qsjs1mLxn7yVxm79YsxWDTgwbj9N5/image.png)

The general laws of exponents follow from our definition together with the law of exponents for e<sup>x</sup> which I covered in my earlier video as well:

If x and y are real numbers and a > 0, then:

1. a<sup>x+y</sup> = a<sup>x</sup>a<sup>y</sup>
2. a<sup>x-y</sup> = a<sup>x</sup>/a<sup>y</sup>
3. (a<sup>x</sup>)<sup>y</sup> = a<sup>xy</sup>
4. (ab)<sup>x</sup> = a<sup>x</sup>b<sup>x</sup>

***I have proven these in later videos.***