Successfully reported this slideshow.
We use your LinkedIn profile and activity data to personalize ads and to show you more relevant ads. You can change your ad preferences anytime.

Logarithms

3,783 views

Published on

  • Be the first to comment

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>

×