Difference between revisions of "Exponential function"
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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 | + | :If <math>b > 1</math>, then the function 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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::[[Image:expfunc_graphsolve_eqn.jpg]] | ::[[Image:expfunc_graphsolve_eqn.jpg]] | ||
| − | *''' | + | *'''Algebraically:''' |
| − | There, | + | There, we will use [[Natural logarithm|natural logarithms]]. The same operation can also be done with [[common logarithms]]. |
::<math>56 = 12\left( {1.24976} \right)^x </math> | ::<math>56 = 12\left( {1.24976} \right)^x </math> | ||
::<math>{{56} \over {12}} = \left( {1.24976} \right)^x </math> | ::<math>{{56} \over {12}} = \left( {1.24976} \right)^x </math> | ||
Latest revision as of 14:57, 6 March 2022
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 function will show growth. - If
, then the function will show decay.
, then the function 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.
- Algebraically:
There, we will use natural logarithms. The same operation can also be done with common logarithms.
