Logarithms

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Logarithms

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

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