7.4 Logarithms p. 499 What   you should learn: Goal 1 Goal 2 Evaluate logarithms Graph logarithmic functions 7.4 Evaluate ...
Evaluating Log Expressions <ul><li>We know 2 2  = 4 and 2 3  = 8  </li></ul><ul><li>But for what value of y does 2 y  = 6?...
Definition of Logarithm to base a <ul><li>Let a & x be positive numbers & a ≠ 1. </li></ul><ul><li>The logarithm of x with...
<ul><li>The definition tells you that the equations log a x = y and a y  = x are equivilant. </li></ul><ul><li>Rewriting f...
Log form   Exp. form <ul><li>log 2 16 = 4 </li></ul><ul><li>log 10 10 = 1 </li></ul><ul><li>log 3 1 = 0 </li></ul><ul><li>...
Evaluate without a calculator <ul><li>log 3 81 =  </li></ul><ul><li>Log 5 125 = </li></ul><ul><li>Log 4 256 = </li></ul><u...
Evaluating logarithms now you try some! <ul><li>Log  4  16 =  </li></ul><ul><li>Log  5  1 = </li></ul><ul><li>Log  4  2 = ...
You should learn the following general forms!!! <ul><li>Log  a  1 = 0  because a 0  = 1 </li></ul><ul><li>Log  a  a = 1  b...
Natural logarithms <ul><li>log  e  x = ln x </li></ul><ul><li>ln  means  log base e </li></ul>
Common logarithms <ul><li>log  10  x = log x </li></ul><ul><li>Understood base 10 if nothing is there. </li></ul>
Common logs and natural logs with a calculator log 10  button ln button
<ul><li>g(x) = log  b  x is the inverse of </li></ul><ul><li>f(x) = b x </li></ul><ul><li>f(g(x)) = x and g(f(x)) = x </li...
<ul><li>So: g(f(x)) =  log b b x  = x </li></ul><ul><li>f(g(x)) =  b log b x  = x </li></ul><ul><li>10 log2  =  </li></ul>...
Finding Inverses <ul><li>Find the inverse of: </li></ul><ul><li>y = log 3 x </li></ul><ul><li>By definition of logarithm, ...
Finding Inverses cont. <ul><li>Find the inverse of : </li></ul><ul><li>Y = ln (x +1) </li></ul><ul><li>X = ln (y + 1)  Swi...
Assignment
Graphs of logs <ul><li>y = log b (x-h)+k  </li></ul><ul><li>Has vertical asymptote x=h </li></ul><ul><li>The domain is x>h...
Graph y =log 5 (x+2) <ul><li>Plot easy points (-1,0) & (3,1) </li></ul><ul><li>Label the asymptote x=-2 </li></ul><ul><li>...
Assignment
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8.4 logarithms1

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8.4 logarithms1

  1. 1. 7.4 Logarithms p. 499 What you should learn: Goal 1 Goal 2 Evaluate logarithms Graph logarithmic functions 7.4 Evaluate Logarithms and Graph Logarithmic Functions A3.2.2
  2. 2. Evaluating Log Expressions <ul><li>We know 2 2 = 4 and 2 3 = 8 </li></ul><ul><li>But for what value of y does 2 y = 6? </li></ul><ul><li>Because 2 2 < 6 < 2 3 you would expect the answer to be between 2 & 3. </li></ul><ul><li>To answer this question exactly, mathematicians defined logarithms. </li></ul>
  3. 3. Definition of Logarithm to base a <ul><li>Let a & x be positive numbers & a ≠ 1. </li></ul><ul><li>The logarithm of x with base a is denoted by log a x and is defined: </li></ul><ul><li>log a x = y iff a y = x </li></ul><ul><li>This expression is read “log base a of x” </li></ul><ul><li>The function f(x) = log a x is the logarithmic function with base a. </li></ul>
  4. 4. <ul><li>The definition tells you that the equations log a x = y and a y = x are equivilant. </li></ul><ul><li>Rewriting forms: </li></ul><ul><li>To evaluate log 3 9 = x ask yourself… </li></ul><ul><li>“ Self… 3 to what power is 9?” </li></ul><ul><li>3 2 = 9 so…… log 3 9 = 2 </li></ul>
  5. 5. Log form Exp. form <ul><li>log 2 16 = 4 </li></ul><ul><li>log 10 10 = 1 </li></ul><ul><li>log 3 1 = 0 </li></ul><ul><li>log 10 .1 = -1 </li></ul><ul><li>log 2 6 ≈ 2.585 </li></ul><ul><li>2 4 = 16 </li></ul><ul><li>10 1 = 10 </li></ul><ul><li>3 0 = 1 </li></ul><ul><li>10 -1 = .1 </li></ul><ul><li>2 2.585 = 6 </li></ul>
  6. 6. Evaluate without a calculator <ul><li>log 3 81 = </li></ul><ul><li>Log 5 125 = </li></ul><ul><li>Log 4 256 = </li></ul><ul><li>Log 2 (1/32) = </li></ul><ul><li>3 x = 81 </li></ul><ul><li>5 x = 125 </li></ul><ul><li>4 x = 256 </li></ul><ul><li>2 x = (1/32) </li></ul>4 3 4 -5
  7. 7. Evaluating logarithms now you try some! <ul><li>Log 4 16 = </li></ul><ul><li>Log 5 1 = </li></ul><ul><li>Log 4 2 = </li></ul><ul><li>Log 3 (-1) = </li></ul><ul><li>(Think of the graph of y=3 x ) </li></ul>2 0 ½ ( because 4 1/2 = 2) undefined
  8. 8. You should learn the following general forms!!! <ul><li>Log a 1 = 0 because a 0 = 1 </li></ul><ul><li>Log a a = 1 because a 1 = a </li></ul><ul><li>Log a a x = x because a x = a x </li></ul>
  9. 9. Natural logarithms <ul><li>log e x = ln x </li></ul><ul><li>ln means log base e </li></ul>
  10. 10. Common logarithms <ul><li>log 10 x = log x </li></ul><ul><li>Understood base 10 if nothing is there. </li></ul>
  11. 11. Common logs and natural logs with a calculator log 10 button ln button
  12. 12. <ul><li>g(x) = log b x is the inverse of </li></ul><ul><li>f(x) = b x </li></ul><ul><li>f(g(x)) = x and g(f(x)) = x </li></ul><ul><li>Exponential and log functions are inverses and “undo” each other </li></ul>
  13. 13. <ul><li>So: g(f(x)) = log b b x = x </li></ul><ul><li>f(g(x)) = b log b x = x </li></ul><ul><li>10 log2 = </li></ul><ul><li>Log 3 9 x = </li></ul><ul><li>10 logx = </li></ul><ul><li>Log 5 125 x = </li></ul>2 Log 3 (3 2 ) x = Log 3 3 2x = 2x x 3x
  14. 14. Finding Inverses <ul><li>Find the inverse of: </li></ul><ul><li>y = log 3 x </li></ul><ul><li>By definition of logarithm, the inverse is y=3 x </li></ul><ul><li>OR write it in exponential form and switch the x & y! 3 y = x 3 x = y </li></ul>
  15. 15. Finding Inverses cont. <ul><li>Find the inverse of : </li></ul><ul><li>Y = ln (x +1) </li></ul><ul><li>X = ln (y + 1) Switch the x & y </li></ul><ul><li>e x = y + 1 Write in exp form </li></ul><ul><li>e x – 1 = y solve for y </li></ul>
  16. 16. Assignment
  17. 17. Graphs of logs <ul><li>y = log b (x-h)+k </li></ul><ul><li>Has vertical asymptote x=h </li></ul><ul><li>The domain is x>h, the range is all reals </li></ul><ul><li>If b>1, the graph moves up to the right </li></ul><ul><li>If 0<b<1, the graph moves down to the right </li></ul>
  18. 18. Graph y =log 5 (x+2) <ul><li>Plot easy points (-1,0) & (3,1) </li></ul><ul><li>Label the asymptote x=-2 </li></ul><ul><li>Connect the dots using the asymptote. </li></ul>X=-2
  19. 19. Assignment
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