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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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----
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== General Info and Definitions ==
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Exponential functions are functions that grows or decays at a constant percent rate.
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:Exponential functions that result in an '''''increase''''' of ''y'' is called an '''''exponential growth'''''.
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:Exponential functions that result in an '''''decrease''''' of ''y'' is called an '''''exponential decay'''''.
  
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Exponential equations are in one of three forms. <math>f\left( x \right) = ab^x </math>, <math>f\left( x \right) = ae^k </math>, where ''e'' is the irrational constant ''2.71828182846....'', or
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An exponential growth graph looks like:
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<math>f\left( x \right) = a\left( {{1 \over 2}} \right)^{{x \over h}}  
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</math> <math>f\left( x \right) = a\left( 2 \right)^{{x \over d}}  
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[[Image:2_power_x_growth.jpg]]
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An exponential decay graph looks like:
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[[Image:05_power_x_decay.jpg]]
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Exponential functions are in one of three forms.  
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:<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 [[e]] is the irrational constant ''2.71828182846....''
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:<math>f\left( x \right) = a\left( {{1 \over 2}} \right)^{{x \over h}}  
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</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.
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:If <math>b > 1</math>, then the function will show growth.
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:If <math>0 < b < 1</math>, then the function will show decay.
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 +
== 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]].
 +
 +
'''Example:''' Solve <math>56 = 12\left( {1.24976} \right)^x </math>
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*'''Graphically:'''
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::Graph both equations and find the intersection.
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::[[Image:expfunc_graphsolve_eqn.jpg]]
  
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''this page is still under construction...more to come very soon''
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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>
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::<math>{{56} \over {12}} = \left( {1.24976} \right)^x </math>
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::<math>\ln \left( {{{56} \over {12}}} \right) = x\ln \left( {1.24976} \right)</math>
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::<math>x = {{\ln \left( {{{56} \over {12}}} \right)} \over {\ln \left( {1.24976} \right)}}</math>
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::<math>x \approx 6.9093</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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