Difference between revisions of "Exponential Functions"

From Department of Mathematics at UTSA
Jump to navigation Jump to search
 
(2 intermediate revisions by the same user not shown)
Line 1: Line 1:
 +
==Operations With Exponential Function==
 +
An exponential function is a function where a constant base (b) is raised to a variable.
 +
 +
===Multiplication===
 +
 +
Firstly, <math>b^x \times b^{2x}</math> is <math>b^{\left(x + 2x\right)}\,</math> which is <math>b^{3x}\,</math>. So when you multiply a base by the same base you add the variables. To clarify, here is an example with numbers:
 +
{| align="center" border=1 cellspacing=0 cellpadding=5
 +
| <math>x\,</math>
 +
| <math>2^x\,</math>
 +
| <math>2^{2x}\,</math>
 +
| <math>2^x \times 2^{2x}\,</math>
 +
| <math>2^{3x}\,</math>
 +
|-
 +
| 1
 +
| 2
 +
| 4
 +
| 8
 +
| 8
 +
|-
 +
| 2
 +
| 4
 +
| 16
 +
| 64
 +
| 64
 +
|}
 +
 +
===Division===
 +
 +
Secondly <math>\frac{b^{2x}}{b^y}</math> is <math>b^{\left( 2x - y \right)}\,</math>. So when a base is divided by the same base you subtract the variables.
 +
 +
Here is an example with numbers:
 +
<math>\frac{2^4}{2^2}=\frac{16}{4}=4=2^2</math>.
 +
 +
===Base raised to two powers===
 +
 +
Thirdly <math>\left(b^{2x}\right)^{3x}</math> is <math>b^{\left(2x\right) \times \left(3x\right)}</math> which is <math>b^{6x^2}</math>. So when a base with a variable is raised to a variable you multiply the variables. Here is another example with numbers: (when x = 1) <math>\left(2^2\right)^3=4^3=64=2^6</math>.
 +
 +
===Multiple bases===
 +
 +
Fourthly when <math>a^2 \times b^2 = ab \times ab</math> it is the same as <math>\left(ab\right)^2\,</math>. Here is an example with numbers: <math>2^2 \times 3^2 = 36 = 6^2</math>. There is a similar situation with division: <math>\left(\frac{a}{b}\right)^2 = \frac{a}{b} \times \frac{a}{b} = \frac{a^2}{b^2}</math>. So when you multiply or divide two different bases raised to the same variable you can multiply or divide them first and then raise them to the variable.
 +
 +
===Fractional exponents===
 +
 +
The last case is when x is presented as a fraction, you can make a square root function, for example <math>b^\frac{1}{x}</math> becomes <math>\pm \sqrt[x] b</math>. However it is customary to only use the positive root and so <math>b^\frac{1}{x}</math> is defined as <math>\sqrt[x] b</math>. Another similar case is when the fraction has a constant (designated as c) other than 1 in the numerator  , for example <math>b^ \frac {3}{x} = \left( \sqrt[x] b \right)^3</math> so <math>b^ \frac {c}{x} = \left( \sqrt[x] b \right)^c</math>.
 +
 +
===The Laws of Exponents===
 +
 +
The rules that have been suggested above are known as the laws of exponents and can be written as:
 +
 +
# <math>b^xb^y = b^{x+y}\,</math>
 +
# <math>\frac{b^x}{b^y} = b^{x-y}</math>
 +
# <math>\left(b^x\right)^y = b^{xy}</math>
 +
# <math>a^n b^n = \left(ab\right)^n\,</math>
 +
# <math>\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}</math>
 +
# <math>b^{-n}=\frac{1}{b^n}</math>
 +
# <math>b^ \frac {c}{x} = \left( \sqrt[x] b \right)^c</math> where c is a constant
 +
# <math>b^1=b\,</math>
 +
# <math>b^0=1\,</math>
 +
 +
===Graphing an Exponential Function===
 +
When you graph an exponential function you use the same methods as with a regular function. There is a graph below that you can look at.
 +
 +
[[Image:Logexponential.svg|right|400px]]
 +
 
== Solving Exponential Equations ==
 
== Solving Exponential Equations ==
  
−
An '''exponential equation''' is an equation in which one or more of the terms is an exponential function. e.g. <math>5^x = 2^{x+2}</math>. Exponential equations can be solved with logarithms.
+
In order to solve an exponential equation you need to make sure that all the bases are the same. Then you can remove the base and solve for the variable. Here is an example:
  
−
e.g. Solve <math>3^{x+1} = 4^{2x-1}</math>
+
Solve for x. <math>2^{\left(x-1\right)} = 16\,</math>
  
−
<math>\begin{align}
+
Now we convert 16 to a base 2 raised to a number.
−
3^{x+1} &= 4^{2x-1} \\
+
 
−
(x+1)\ln 3 &= (2x-1)\ln 4 \\
+
<math>2^{\left(x-1\right)} = 2^4\,</math>
−
x\ln 3 + \ln 3 &= 2x\ln 4 - \ln 4 \\
+
 
−
\ln 3 + \ln 4 &= x(2\ln 4 - \ln 3) \\
+
Now we can remove the base. So we have:
−
x &= \frac{\ln 3 + \ln 4}{2\ln 4 - \ln 3} \\
+
 
−
x &\approx 1.4844
+
<math>x-1 = 4\,</math>
−
\end{align}</math>
+
 
 +
Finally solve for x.
 +
 
 +
<math>x = 5\,</math>
  
 
==Resources==
 
==Resources==
 
* [https://mathresearch.utsa.edu/wikiFiles/MAT1053/Exponential%20Functions/MAT1053_M5.1Exponential_Functions.pdf Exponential Functions], Book Chapter
 
* [https://mathresearch.utsa.edu/wikiFiles/MAT1053/Exponential%20Functions/MAT1053_M5.1Exponential_Functions.pdf Exponential Functions], Book Chapter
 
* [https://mathresearch.utsa.edu/wikiFiles/MAT1053/Exponential%20Functions/MAT1053_M5.1Exponential_FunctionsGN.pdf Guided Notes]
 
* [https://mathresearch.utsa.edu/wikiFiles/MAT1053/Exponential%20Functions/MAT1053_M5.1Exponential_FunctionsGN.pdf Guided Notes]
 +
 +
== Licensing ==
 +
Content obtained and/or adapted from:
 +
* [https://en.wikibooks.org/wiki/A-level_Mathematics/OCR/C2/Logarithms_and_Exponentials Logarithms and Exponentials, Wikibooks: A-level Mathematics/OCR/C2] under a CC BY-SA license

Latest revision as of 14:44, 21 October 2021

Operations With Exponential Function

An exponential function is a function where a constant base (b) is raised to a variable.

Multiplication

Firstly, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b^x \times b^{2x}} is which is . So when you multiply a base by the same base you add the variables. To clarify, here is an example with numbers:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x\,} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2^x\,} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2^{2x}\,} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2^x \times 2^{2x}\,} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2^{3x}\,}
1 2 4 8 8
2 4 16 64 64

Division

Secondly Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{b^{2x}}{b^y}} is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b^{\left( 2x - y \right)}\,} . So when a base is divided by the same base you subtract the variables.

Here is an example with numbers: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{2^4}{2^2}=\frac{16}{4}=4=2^2} .

Base raised to two powers

Thirdly Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left(b^{2x}\right)^{3x}} is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b^{\left(2x\right) \times \left(3x\right)}} which is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b^{6x^2}} . So when a base with a variable is raised to a variable you multiply the variables. Here is another example with numbers: (when x = 1) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left(2^2\right)^3=4^3=64=2^6} .

Multiple bases

Fourthly when Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a^2 \times b^2 = ab \times ab} it is the same as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left(ab\right)^2\,} . Here is an example with numbers: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2^2 \times 3^2 = 36 = 6^2} . There is a similar situation with division: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left(\frac{a}{b}\right)^2 = \frac{a}{b} \times \frac{a}{b} = \frac{a^2}{b^2}} . So when you multiply or divide two different bases raised to the same variable you can multiply or divide them first and then raise them to the variable.

Fractional exponents

The last case is when x is presented as a fraction, you can make a square root function, for example Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b^\frac{1}{x}} becomes Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pm \sqrt[x] b} . However it is customary to only use the positive root and so Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b^\frac{1}{x}} is defined as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sqrt[x] b} . Another similar case is when the fraction has a constant (designated as c) other than 1 in the numerator , for example Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b^ \frac {3}{x} = \left( \sqrt[x] b \right)^3} so Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b^ \frac {c}{x} = \left( \sqrt[x] b \right)^c} .

The Laws of Exponents

The rules that have been suggested above are known as the laws of exponents and can be written as:

  1. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b^xb^y = b^{x+y}\,}
  2. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{b^x}{b^y} = b^{x-y}}
  3. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left(b^x\right)^y = b^{xy}}
  4. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a^n b^n = \left(ab\right)^n\,}
  5. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}}
  6. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b^{-n}=\frac{1}{b^n}}
  7. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b^ \frac {c}{x} = \left( \sqrt[x] b \right)^c} where c is a constant
  8. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b^1=b\,}
  9. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b^0=1\,}

Graphing an Exponential Function

When you graph an exponential function you use the same methods as with a regular function. There is a graph below that you can look at.

Logexponential.svg

Solving Exponential Equations

In order to solve an exponential equation you need to make sure that all the bases are the same. Then you can remove the base and solve for the variable. Here is an example:

Solve for x. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2^{\left(x-1\right)} = 16\,}

Now we convert 16 to a base 2 raised to a number.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2^{\left(x-1\right)} = 2^4\,}

Now we can remove the base. So we have:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x-1 = 4\,}

Finally solve for x.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x = 5\,}

Resources

Licensing

Content obtained and/or adapted from:

Retrieved from "https://mathresearch.utsa.edu/wiki/index.php?title=Exponential_Functions&oldid=2811"