Difference between revisions of "Exponential function"

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:

An exponential decay graph looks like:

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.

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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-#REDIRECT [[Exponents]]
+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]].
+== 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:
+[[Image:2_power_x_growth.jpg]]
+An exponential decay graph looks like:
+[[Image:05_power_x_decay.jpg]]
+Exponential functions are in one of three forms.
+:<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....''
+:<math>f\left( x \right) = a\left( {{1 \over 2}} \right)^{{x \over h}}
+</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).
+Whether an exponential function shows growth or decay depends upon the value of its ''b'' value.
+:If <math>b > 1</math>, then the function will show growth.
+:If <math>0 < b < 1</math>, 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 <math>56 = 12\left( {1.24976} \right)^x </math>
+*'''Graphically:'''
+::Graph both equations and find the intersection.
+::[[Image:expfunc_graphsolve_eqn.jpg]]
+*'''Algebraically:'''
+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} \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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Difference between revisions of "Exponential function"

Latest revision as of 14:57, 6 March 2022

General Info and Definitions

Solving Exponential Equations