1 / 20

Understanding Logarithms: Definitions, Evaluations, and Inverses

This guide delves into the fundamental concepts of logarithms, including their definitions, evaluations, and properties concerning their inverses. It explains the logarithmic relationship, provides examples of calculating logarithmic values, and discusses natural and common logarithms. Additionally, it addresses the graphing of logarithmic functions and their asymptotic behavior, demonstrating how these mathematical concepts interrelate. This comprehensive overview aims to enhance your understanding of logarithmic expressions and their practical applications.

cameron-rush
Télécharger la présentation

Understanding Logarithms: Definitions, Evaluations, and Inverses

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. 8.4 Logarithms By: L. Keali’i Alicea

  2. Evaluating Log Expressions • We know 22 = 4 and 23 = 8 • But for what value of y does 2y = 6? • Because 22<6<23 you would expect the answer to be between 2 & 3. • To answer this question exactly, mathematicians defined logarithms.

  3. Definition of Logarithm to base a • Let a & x be positive numbers & a ≠ 1. • The logarithm of x with base a is denoted by logax and is defined: • logax = y iff ay = x • This expression is read “log base a of x” • The function f(x) = logax is the logarithmic function with base a.

  4. The definition tells you that the equations logax = y and ay = x are equivilant. • Rewriting forms: • To evaluate log3 9 = x ask yourself… • “Self… 3 to what power is 9?” • 32 = 9 so…… log39 = 2

  5. log216 = 4 log1010 = 1 log31 = 0 log10 .1 = -1 log2 6 ≈ 2.585 24 = 16 101 = 10 30 = 1 10-1 = .1 22.585 = 6 Log formExp. form

  6. log381 = Log5125 = Log4256 = Log2(1/32) = 3x = 81 5x = 125 4x = 256 2x = (1/32) Evaluate without a calculator 4 3 4 -5

  7. Evaluating logarithms now you try some! 2 • Log 4 16 = • Log 5 1 = • Log 4 2 = • Log 3 (-1) = • (Think of the graph of y=3x) 0 ½ (because 41/2 = 2) undefined

  8. You should learn the following general forms!!! • Log a 1 = 0 because a0 = 1 • Log a a = 1 because a1 = a • Log a ax = x because ax = ax

  9. Natural logarithms • log e x = ln x • ln means log base e

  10. Common logarithms • log 10 x = log x • Understood base 10 if nothing is there.

  11. Common logs and natural logs with a calculator log10 button ln button

  12. g(x) = log b x is the inverse of • f(x) = bx • f(g(x)) = x and g(f(x)) = x • Exponential and log functions are inverses and “undo” each other

  13. So: g(f(x)) = logbbx = x • f(g(x)) = blogbx = x • 10log2 = • Log39x = • 10logx = • Log5125x = 2 Log3(32)x = Log332x= 2x x 3x

  14. Finding Inverses • Find the inverse of: • y = log3x • By definition of logarithm, the inverse is y=3x • OR write it in exponential form and switch the x & y! 3y = x 3x = y

  15. Finding Inverses cont. • Find the inverse of : • Y = ln (x +1) • X = ln (y + 1) Switch the x & y • ex = y + 1 Write in exp form • ex – 1 = y solve for y

  16. Assignment

  17. Graphs of logs • y = logb(x-h)+k • Has vertical asymptote x=h • The domain is x>h, the range is all reals • If b>1, the graph moves up to the right • If 0<b<1, the graph moves down to the right

  18. Graph y = log1/3x-1 • Plot (1/3,0) & (3,-2) • Vert line x=0 is asy. • Connect the dots X=0

  19. Graph y =log5(x+2) • Plot easy points (-1,0) & (3,1) • Label the asymptote x=-2 • Connect the dots using the asymptote. X=-2

  20. Assignment 8.4 A (all) 8.4 B (1-27 odd, 28-29)

More Related