1 / 18

Lesson 81

Lesson 81. Using natural logarithms. Natural logarithms. When the base is e , the logarithm is called a natural logarithm e x = a means ln a= x. Inverse properties of logarithms. e ln x = x and ln e x = x where x>0.

libba
Download Presentation

Lesson 81

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Lesson 81 Using natural logarithms

  2. Natural logarithms • When the base is e, the logarithm is called a natural logarithm • ex = a means ln a= x

  3. Inverse properties of logarithms • eln x = x and ln ex = x • where x>0

  4. Simplifying exponential and logarithmic expressions • Simplify elnp • elnp = p • Simplify ln e3c2+1= • = 3c2+1 • simplfy eln2x • simplify ln e2d2+d

  5. Properties of natural logarithms • Product property • ln ab = ln a + ln b • Quotient Property • ln (a/b) = ln a - ln b • Power Property • ln ap = p ln a

  6. Applying properties of natural logarithms • Rewrite as a sum or difference • ln 6e= ln 6 + ln e • = ln 6 + 1 • ln 3e = ln 3e- ln x=ln3 + ln e-ln x • x • =ln 3 +1 - lnx • ln e5x2 = 5x2 ln e = 5x2 (1) = 5x2

  7. practice • Write as a sum or difference- then simplify • ln 10e • ln 5e • y • 3 ln e6x2 • ln( 4c3)4 • e2

  8. Continuous exponential growth • The formula for exponential growth where interest is compounded continuously is • A = P ert

  9. Lesson 87 • Evaluating logarithmic expressions

  10. Evaluating expressions of the form loga(bc)d • Use the properties of logs to evaluate • log4(16x)3 when x = 256 • =3log4(16x) • =3(log416+log4x) • Since log416 = 2 • = 3(2+log4x) • = 6 +3log4x • When x = 256 • = 6+ 3log4 256= 6 + 3 (4)= 18

  11. evaluate • ln(7e)2 • = 2 ln(7e) • =2 (ln 7+ln e) • = 2(ln 7 + 1) • = 2 ln 7 + 2 • = 2(1.95) + 2 = 5.9

  12. practice • Evaluate when r = 243 • log3(27r)4 • Evaluate • ln(12e)3

  13. the change of base formula • remember the change of base formula where a is the new base

  14. using the change of base formula • Convert log100(10x)2 to base 10. then evaluate when x = 1000 • =2 log100 (10x) • =2( log 10x) • log 100 • = 2 (log10 +logx) • log100 • =2( 1 + log 1000)= 2( 1+3) = 4 • 2 2

  15. use change of base formula • Convert to base e, then evaluate when x = 6 • log4(2x)3= 3 log4(2x) • =3 (ln 2x)= 3 (ln 2 + ln x) • ln 4 ln 4 • = 3 ( ln2 + ln 6) • ln 4 • = 3( .6931 + 1.7918) = 5.3774 • 1.3863

  16. practice • solve - change to base 10 • log 100(1000x)3 when x = 10 • change to base e • log9(3x)5 when x = 4

  17. Solving log equations using the change of base formula • Solve 1000 9x = 100 • log 1000 100 = 9x • change to base 10 • log 100 = 9x • log 1000 • 2 = 9x • 3 • 2 = 27x x = 2/27

  18. practice • Solve for x: • 32 4x = 8

More Related