Difference between revisions of "Exponential function"
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− | An exponential growth graph looks like: | + | An exponential growth graph looks like: [[Image:2_power_x_growth.jpg]] |
− | [[Image:2_power_x_growth.jpg]] | ||
− | An exponential decay graph looks like: | + | An exponential decay graph looks like: [[Image:05_power_x_decay.jpg]] |
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'''Example:''' Solve <math>56 = 12\left( {1.24976} \right)^x </math> | '''Example:''' Solve <math>56 = 12\left( {1.24976} \right)^x </math> | ||
− | '''Graphically:''' | + | *'''Graphically:''' |
+ | |||
+ | ::Graph both equations and find the intersection. | ||
+ | ::[[Image:expfunc_graphsolve_eqn.jpg]] | ||
+ | |||
+ | *'''Algebraicly:''' | ||
+ | ::<math>56 = 12\left( {1.24976} \right)^x </math> | ||
+ | ::<math>{{56} \over {12}} = \left( {1.24976} \right)^x </math> | ||
+ | ::<math>\ln \left( {{{56} \over {12}}} \right) = x\ln \left( {1.24976} \right)</math> | ||
+ | ::<math>x = {{\ln \left( {{{56} \over {12}}} \right)} \over {\ln \left( {1.24976} \right)}}</math> | ||
+ | ::<math>x \approx 6.9093</math> | ||
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Revision as of 09:31, 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).
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.
Solving Exponential Equations
There are two ways to solve an exponential equation. Graphically with a computer/calculator or algebraicly using logarithms.
Example: Solve
- Graphically:
- Algebraicly:
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