Difference between revisions of "Exponential function"
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The '''exponential function''' is the [[function]] <math>f(x) = e^x</math>, [[exponentiation]] by ''[[e]]''. It is a very important function in [[analysis]], both [[real]] and [[complex]]. | The '''exponential function''' is the [[function]] <math>f(x) = e^x</math>, [[exponentiation]] by ''[[e]]''. It is a very important function in [[analysis]], both [[real]] and [[complex]]. | ||
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== General Info and Definitions == | == General Info and Definitions == | ||
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Exponential functions are functions that grows or decays at a constant percent rate. | 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 '''''increase''''' of ''y'' is called an '''''exponential growth'''''. | ||
:Exponential functions that result in an '''''decrease''''' of ''y'' is called an '''''exponential decay'''''. | :Exponential functions that result in an '''''decrease''''' of ''y'' is called an '''''exponential decay'''''. | ||
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An exponential growth graph looks like: | An exponential growth graph looks like: | ||
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[[Image:05_power_x_decay.jpg]] | [[Image:05_power_x_decay.jpg]] | ||
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Exponential functions are in one of three forms. | Exponential functions are in one of three forms. | ||
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</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}} | ||
</math>, where ''h'' is the half-life (for decay), or ''d'' is the doubling time (for growth). | </math>, where ''h'' is the half-life (for decay), or ''d'' is the doubling time (for growth). | ||
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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. | ||
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== Solving Exponential Equations == | == Solving Exponential Equations == | ||
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There are two ways to solve an exponential equation. Graphically with a computer/calculator or algebraicly using [[logarithms]]. | There are two ways to solve an exponential equation. Graphically with a computer/calculator or algebraicly using [[logarithms]]. | ||
Revision as of 15:43, 19 April 2008
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:
There, I will use natural logarithms. The same opperation can also be done with common logarithms.
