Difference between revisions of "Math 22 Exponential Functions"

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7.<math>a^{-x}=\frac{1}{a^x}</math>
 
7.<math>a^{-x}=\frac{1}{a^x}</math>
  
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'''Exercises''' Use the properties of exponents to simplify each expression:
  
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'''a)''' <math>(8^{\frac{1}{2}})(2^{\frac{1}{2}})</math>
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{| class = "mw-collapsible mw-collapsed" style = "text-align:left;"
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!Solution: &nbsp;
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|-
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|<math>(8^{\frac{1}{2}})(2^{\frac{1}{2}})=(8\cdot 2)^{\frac{1}{2}}=16^{\frac{1}{2}}</math>
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|}
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'''b)''' <math>\frac{7^5}{49^3}</math>
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{| class = "mw-collapsible mw-collapsed" style = "text-align:left;"
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!Solution: &nbsp;
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|-
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|<math>\frac{7^5}{49^3}=\frac{7^5}{(7^2)^3}=\frac{7^5}{7^{2.3}}=\frac{7^5}{7^6}=7^{5-6}=7^{-1}=\frac{1}{7}</math>
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|}
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'''c)''' <math>(\frac{1}{4})^2(4^2)</math>
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{| class = "mw-collapsible mw-collapsed" style = "text-align:left;"
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!Solution: &nbsp;
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|-
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|<math>(\frac{1}{4})^2(4^2)=(4^{-2})(4^2)=4^{-2+2}=4^0=1</math>
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|}
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==Graphs of Exponential Functions==
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The graph of the exponential function <math>a^x</math> where <math>a>0, a\ne 1</math> always goes through the point <math>(0,1)</math> and has a horizontal asymptote <math>y=0</math>
  
 
[[Math_22| '''Return to Topics Page''']]
 
[[Math_22| '''Return to Topics Page''']]
  
 
'''This page were made by [[Contributors|Tri Phan]]'''
 
'''This page were made by [[Contributors|Tri Phan]]'''

Latest revision as of 07:44, 11 August 2020

Definition of Exponential Function

 If  and , then the exponential function with base  is

Properties of Exponents

Let and be positive real numbers, and let 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} and 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 y} be real numbers.

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 a^0=1}

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 a^xa^y=a^{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 \frac{a^x}{a^y}=a^{x-y}}

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^x)^y=a^{xy}}

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 (ab)^x=a^xb^x}

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 (\frac{a}{b})^x=\frac{a^x}{b^x}}

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 a^{-x}=\frac{1}{a^x}}

Exercises Use the properties of exponents to simplify each expression:

a) 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 (8^{\frac{1}{2}})(2^{\frac{1}{2}})}

Solution:  
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 (8^{\frac{1}{2}})(2^{\frac{1}{2}})=(8\cdot 2)^{\frac{1}{2}}=16^{\frac{1}{2}}}

b) 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{7^5}{49^3}}

Solution:  
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{7^5}{49^3}=\frac{7^5}{(7^2)^3}=\frac{7^5}{7^{2.3}}=\frac{7^5}{7^6}=7^{5-6}=7^{-1}=\frac{1}{7}}

c) 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{1}{4})^2(4^2)}

Solution:  
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{1}{4})^2(4^2)=(4^{-2})(4^2)=4^{-2+2}=4^0=1}

Graphs of Exponential Functions

The graph of the exponential function 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^x} where 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>0, a\ne 1} always goes through the point 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 (0,1)} and has a horizontal asymptote 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 y=0}

Return to Topics Page

This page were made by Tri Phan

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