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
If a > 0 and r is any rational number then by the laws I covered in my earlier videos:
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):
The function f(x) = ax is called the exponential function with base a.
Notice that ax is positive because ex is positive.
This definition allows to us to extend another law that I already showed in my earlier video:
The general laws of exponents follow from our definition together with the law of exponents for ex which I covered in my earlier video as well:
If x and y are real numbers and a > 0, then:
- ax+y = axay
- ax-y = ax/ay
- (ax)y = axy
- (ab)x = axbx
I have proven these in later videos.
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