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The Natural Base , e , is defined to be

Notes Day 8.4 PAP Algebra 2 Objective: TLW… develop the study of logarithms by exploring the Natural base e and Natural logarithms ln. The Natural Base , e , is defined to be. Irrational. The number e is ________________ It is also known as Euler’s number. Domain: Range:

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The Natural Base , e , is defined to be

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  1. Notes Day 8.4 PAP Algebra 2Objective: TLW…develop the study of logarithms by exploring the Natural base e and Natural logarithms ln

  2. TheNatural Base, e , is defined to be Irrational The number e is ________________ It is also known as Euler’s number

  3. Domain: Range: Y-intercept: Asymptote: Graph f(x)= ex Complete a t-chart All reals x Y>0 Type of Function:: (0,1) Growth Y=0

  4. Domain: Range: Y-intercept: Asymptote: Graph f(x)= e-x Complete a t-chart Decay All reals x Type of Function:: Y>0 f(x) =eax, rep exp growth when a is ___________ and decay when a is _________ (0,1) Positive Y=0 negative

  5. SOLVE for x in the following Use the calculator and round 3 places

  6. Change the following exponential fcns to logarithmic form 1. 2. So lnx = 2 In the next slide you will see why this can also be written as a natural log

  7. Natural Logarithms If x is a positive real number, then the natural logarithm of x is denoted by: or Note that if a base is not written here – it is base e

  8. Calculator • Calculators can evaluate logs with the common base….. which is base 10 • They can also evaluate the natural logarithm…which is base e , the natural number Use the log key Use the ln key

  9. These 2 graphs are reflections over the line _________ f(x)= ex f(x)= ln x inverse Is ________ of f(x)= logex Inverses Exp fcn Natural log fcn HA: y=o VA: x=o y = x

  10. Special Values of Logarithms 1 0 2 ½ ln 1=___ ln e=___ ln e2 =___ ln e1/2 =___ X=1 X=2 X=0 X=½ ln e cancels to equal 1

  11. Review: Properties of Exponents v Properties of Logarithms this is a review and not on your notes PRODUCT QUOTIENT POWER

  12. SOLVE for x using the properties of logs ln 12 = ln 3 + ln x

  13. SOLVE for x using the properties of logs ln 8 = 3 ln x

  14. SOLVE for x using the properties of logs ln 8 – ln 32 = ln x

  15. Sketch:f(x) = 2 – ln x Can also be written as: f(x) = – ln x + 2 This is a log fcn with base e and is to be shifted: The negative causes a reflection in the x axis. The 2 cause a vertical shift up 2 Remember: ln 1 = 0 Log fcn has VA: no horizontal shift so x=o

  16. Activity: Now lets see what you know. Work the following on a marker board. You will also need a calculator. TEST: next class HW : WS 8.4 – which is due next class

  17. Solve for x with no calculator: 1 It is helpful to memorize ln e = 1

  18. Solve for x with no calculator: 0 It is helpful to memorize ln1=0

  19. Solve with a Calculator! X=0.882

  20. Solve with a Calculator! Or more commonly seen Take ln of both sides To eliminate the base e Use the change of base formula X=0.693

  21. Solve with a Calculator! X=-2.996 Take ln of both sides To eliminate the base e

  22. Solve with a Calculator! Convert to a log Use the change of base formula X=3.585 Does this equal Log 6 NO!

  23. Solve with a Calculator! X=3.975

  24. Solve with a Calculator! Or more commonly seen Convert to a log Convert to an exponential X=23.728

  25. Solve with a Calculator! or X=7.389

  26. Solve! X=18

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