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# Logarithms

## by srobbins4 on Jul 23, 2010

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## LogarithmsPresentation Transcript

• Logarithms One-to-One Functions Definition Evaluate Properties
• One-to-one functions
• Functions which have 1-to-1 correspondence:
• every x and y pair are unique
• every x has one corresponding y
• every y has one corresponding x
• no x repeats & no y repeats
• How to determine if functions are 1-to-1:
• Does every element of the domain ( x ) correspond to one and only one member in the range ( y )?
• Examples & Non-examples
• 1-to-1 Functions Not 1-to-1 Functions
• {(0,2), (-3,1), (4,5)} {(0,2), (-3,2), (0,5)}
0 11 4 1 3 -6 -2 5 y x -3 -5 4 1 0 -2 7 -8 0 1 -3 -5 4 1 0 -2 7 -8 0 1 0 11 -2 1 3 5 -2 5 y x
• Inverse Functions
• Inverse: Interchange the x and y values of all ordered pairs (domain and range) 1-to-1 Functions always have inverses that are also functions
• Function Inverse Function
• {(0,2), (-3,1), (4,5)} {(2,0), (1,-3), (5,4)}
0 11 4 1 3 -6 -2 5 y x 11 0 1 4 -6 3 5 -2 y x
• Practice
• Are the following functions one-to-one?
• 1. {(2,7), (1,-1), (2,8)} 3.
• 2.
• **use calculator
• Find the inverse of each function.
• 4. {(-6,3), (9,-1)} 5.
-2 5 5 -2 2 -5 y x
• Logarithmic Functions
• The inverse of an exponential function
• Exponential Logarithmic
• function function
2 1 1 0 1/2 -1 y x 1 2 0 1 -1 1/2 y x
• Graphs of Logarithmic Functions
• Make a table of values, then use transformations to shift the graph
• Vertical asymptote x =0
• Domain: x > 0
• Range: All real numbers
1 b 0 1 -1 1/b y x
• Graph Transformations
• Describe how the graph changes from y=ln x :
• 1.
• 2.
• Logarithms
• To calculate a logarithm, you should convert it to exponential form
• **logarithm = the exponent on the base
• Logarithmic form Exponential form
Base Exponent
• Special Bases
• Common logarithms = base 10
• Natural logarithms = base e
• Practice Conversions
• Rewrite the expression in exponential form
• Rewrite the expression in logarithmic form
• Evaluate each logarithm
• Laws/Properties of Exponents Review ( a is a nonzero real number)
•  leave the same base, add exponents
•  leave the same base, subtract exponents
•  leave the same base, multiply exponents
•  raise each base to the exponent
• outside parentheses
•  flip the base and make the exponent
• positive (find the reciprocal)
•  any base to the zero power = 1
• Properties of Logarithms
• Practice
• Expand using the properties of exponents
• 1.
• 2.
• Practice
• Condense into a single logarithm using the properties of logarithms
• 1.
• 2.
• What’s up next?
• Thursday at 10 AM:
• Solving Logarithmic Equations
• Solving Exponential Equations using Logarithms