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==Rules of Exponents== If s, t, a, b are real numbers with a, b $>$ 0, then $a^s\cdot a^t = a^{s+t} ~ (a^s)^t = a^{..."$

2015-10-23T04:28:56Z

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==Rules of Exponents==

If s, t, a, b are real numbers with a, b <math> > </math> 0, then
<math>a^s\cdot a^t = a^{s+t} ~ (a^s)^t = a^{st}~(ab)^s = a^sb^s
1^s = 1~a^{-s} = \frac{1}{a^s} = \left(\frac{1}{a}\right)^s~a^0 = 1</math>

Now that we can define an exponential function:
<math>f(x) = Ca^x</math> where a is a positive number, that is not 1, and C is a nonzero number.
Then f(x) is an exponential function. We call c the initial value, because if x is a variable for time,
f(0) = C.

==Properties==

The first thing we note, is if <math>f(x) = Ca^x</math> is an exponential function,

then <math>\frac{f(x+1)}{f(x)} = a</math>

==Properties of the graph==

Properties of <math>f(x) = a^x,~ a > 1</math>
1. The domain is <math>(-\infty, \infty)</math> and the range is <math>(0, \infty)</math>
2. The y-intercept is (0, 1) and there is no x-intercept.
3. The x-axis is a horizontal asymptote
4. <math> f(x)</math> is an increasing, one-to-one function
5. The graph contains the three points <math>(0, 1),~(1, a),~(-1, \frac{1}{a})</math>
6. The graph of f is smooth and continuous. (Here smooth means you can take as many derivatives as you want)

Note: You do not have to worry about what it means for a function to be smooth, or what a derivative is, until calculus.

Properties of <math>f(x) = a^x, ~ 0 < a < 1 </math>
1. This type of exponential function has the same properties as the one above EXCEPT in property 4, f(x) is decreasing instead of increasing.

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Exponential Functions - Revision history

MathAdmin: Created page with "__TOC__ ==Rules of Exponents== If s, t, a, b are real numbers with a, b > 0, then a^s\cdot a^t = a^{s+t} ~ (a^s)^t = a^{..."

MathAdmin: Created page with "
TOC
==Rules of Exponents== If s, t, a, b are real numbers with a, b $>$ 0, then $a^s\cdot a^t = a^{s+t} ~ (a^s)^t = a^{..."$