Difference between revisions of "Exponential function"
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Exponential functions are in one of three forms. | Exponential functions are in one of three forms. | ||
:<math>f\left( x \right) = ab^x </math>, where ''b'' is the % change written in decimals | :<math>f\left( x \right) = ab^x </math>, where ''b'' is the % change written in decimals | ||
| − | :<math>f\left( x \right) = ae^k </math>, where [[ | + | :<math>f\left( x \right) = ae^k </math>, where [[e]] is the irrational constant ''2.71828182846....'' |
:<math>f\left( x \right) = a\left( {{1 \over 2}} \right)^{{x \over h}} | :<math>f\left( x \right) = a\left( {{1 \over 2}} \right)^{{x \over h}} | ||
</math> or <math>f\left( x \right) = a\left( 2 \right)^{{x \over d}} | </math> or <math>f\left( x \right) = a\left( 2 \right)^{{x \over d}} | ||
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Whether an exponential function shows growth or decay depends upon the value of its ''b'' value. | Whether an exponential function shows growth or decay depends upon the value of its ''b'' value. | ||
:If <math>b > 1</math>, then the funciton will show growth. | :If <math>b > 1</math>, then the funciton will show growth. | ||
| − | :If <math>0 < b < 1</math>, then the function will show decay. | + | :If <math>0 < b < 1</math>, then the function will show decay. |
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== Solving Exponential Equations == | == Solving Exponential Equations == | ||
Revision as of 10:35, 10 November 2006
The exponential function is the function 
, exponentiation by e. It is a very important function in analysis, both real and complex.
General Info and Definitions
Exponential functions are functions that grows or decays at a constant percent rate.
- Exponential functions that result in an increase of y is called an exponential growth.
- Exponential functions that result in an decrease of y is called an exponential decay.
An exponential growth graph looks like:
An exponential decay graph looks like:
Exponential functions are in one of three forms.
, where b is the % change written in decimals
, where e is the irrational constant 2.71828182846....
or 
, where h is the half-life (for decay), or d is the doubling time (for growth).
, where b is the % change written in decimals
, where 
or 
, where h is the half-life (for decay), or d is the doubling time (for growth).
Whether an exponential function shows growth or decay depends upon the value of its b value.
- If
, then the funciton will show growth. - If
, then the function will show decay.
, then the funciton will show growth.
, then the function will show decay.Solving Exponential Equations
There are two ways to solve an exponential equation. Graphically with a computer/calculator or algebraicly using logarithms.
Example: Solve 
- Graphically:
- Graph both equations and find the intersection.
- Algebraicly:
