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A Journey with

A Journey with. Exponential &. Functions. Logarithmic. Slide of 20. Exponentials and Logarithms. Log Laws. Natural logs. Logarithmic and Exponential Equations. Slide of 20. Exponentials and Logarithms :. y = b 2 (where b > 0). is an exponential function.

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A Journey with

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  1. A Journey with Exponential & Functions Logarithmic Slide of 20

  2. Exponentials and Logarithms • Log Laws • Natural logs • Logarithmic and Exponential Equations. Slide of 20

  3. Exponentials and Logarithms : y = b2(where b > 0) is an exponential function y = logb x(where b > 0 and b  1) is a logarithmic function One function is the inverse of the other. y = bx x = logb y Slide of 20

  4. Examples: (a)23 = 8may be written as log2 8 = 3 (b) may be written as (c)a1 = amay be written asloga a = 1 (d)a0 = 1may be written asloga 1 = 0 Slide of 20

  5. Example: Evaluate log8 16 Let log8 16 = x 16 = 8x 24 = (23)x 24 = 23x 4 = 3x Slide of 20

  6. y Graph of y = 2x 8 6 4 2 x -4 -2 2 4 x axis is an asymptote Slide of 20

  7. Graph of Or y = 2-x y 8 6 4 2 x -4 -2 2 4 Slide of 20

  8. Graph of y = log3 x y 3 1 -1 -3 x 1 3 5 7 9 Slide of 20

  9. A special exponential function is y = ex ( where e 2.718 ) Its inverse is loge x which is written ln x Slide of 20

  10. Log Laws log ab = log a + log b log a - log b log a x = x log a loga 1 = 0 loga a = 1 Slide of 20

  11. Examples: (1) loga 4 + loga 3 = loga (4  3) = loga 12 (2) 2 logb 6 = logb 62 = logb 36 Slide of 20

  12. Examples: (3) loga 12 - 3 loga 2 = loga 12 - loga 23 = loga 12 - loga 8 = loga 1.5 Slide of 20

  13. Examples: (4) log5 25 OR log5 25 Let log5 25 = x = log5 52 25 = 5x = 2 log5 5 52 = 5x = 2  1 x = 2 = 2 Slide of 20

  14. Examples: Simplify loga xy - 3 loga y = loga xy - loga y3 Slide of 20

  15. Natural logs Logs to base 10 are called common logs. Logs to base e are called natural logs. Log10 a is written log 10 Loge a is written ln a To calculate logs to any other base (x) use the rule: [note: your calculator uses base 10 for log x ] Slide of 20

  16. Examples: ln 5 log 5 (1) log6 5 = or log 6 ln 6 = 0.9 = 0.9 (2) ln x = 6 x = e6  403 Slide of 20

  17. Examples: Solve for y: ln y - ln x2 = 3 e3 20 y  20x2 Slide of 20

  18. Logarithmic and Exponential Equations. Given ax = b to solve for x, use logs. x = log4 7 (1) 4x = 7 or log 4x = log 7 x log 4 = log 7 [check: 41.4 = 7] Slide of 20

  19. (2) 32x-4 = 5 2x - 4 = log3 5 2x - 4  1.46 2x  5.46 x  2.73 [check: 3(22.73 - 4) = 5] Slide of 20

  20. The End. Slide of 20

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