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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:
  
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An exponential growth graph looks like:
 
 
[[Image:2_power_x_growth.jpg]]
 
[[Image:2_power_x_growth.jpg]]
  
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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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:If <math>b > 1</math>, then the funciton will show growth.  
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: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.
  
 
== 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]].  
  
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::[[Image:expfunc_graphsolve_eqn.jpg]]
 
::[[Image:expfunc_graphsolve_eqn.jpg]]
  
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*'''Algebraicly:'''
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*'''Algebraically:'''
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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 $f(x) = e^x$, 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:

2 power x growth.jpg

An exponential decay graph looks like:

05 power x decay.jpg

Exponential functions are in one of three forms.

$f\left( x \right) = ab^x$, where b is the % change written in decimals
$f\left( x \right) = ae^k$, where e is the irrational constant 2.71828182846....
$f\left( x \right) = a\left( {{1 \over 2}} \right)^{{x \over h}}$ or $f\left( x \right) = a\left( 2 \right)^{{x \over d}}$, 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 $b > 1$, then the function will show growth.
If $0 < b < 1$, 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 $56 = 12\left( {1.24976} \right)^x$

  • Graphically:
Graph both equations and find the intersection.
Expfunc graphsolve eqn.jpg
  • Algebraically:

There, we will use natural logarithms. The same operation can also be done with common logarithms.

$56 = 12\left( {1.24976} \right)^x$
${{56} \over {12}} = \left( {1.24976} \right)^x$
$\ln \left( {{{56} \over {12}}} \right) = x\ln \left( {1.24976} \right)$
$x = {{\ln \left( {{{56} \over {12}}} \right)} \over {\ln \left( {1.24976} \right)}}$
$x \approx 6.9093$
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