8.4 Logarithms

download 8.4 Logarithms

of 20

  • date post

    01-Jan-2016
  • Category

    Documents

  • view

    25
  • download

    2

Embed Size (px)

description

8.4 Logarithms. p. 486. Evaluating Log Expressions. We know 2 2 = 4 and 2 3 = 8 But for what value of y does 2 y = 6? Because 2 2

Transcript of 8.4 Logarithms

  • 8.4 Logarithmsp. 486

  • Evaluating Log ExpressionsWe know 22 = 4 and 23 = 8 But for what value of y does 2y = 6?Because 22
  • Definition of Logarithm to base aLet 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 = xThis expression is read log base a of xThe function f(x) = logax is the logarithmic function with base a.

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

  • Log form Exp. formlog216 = 4log1010 = 1log31 = 0log10 .1 = -1log2 6 2.58524 = 16101 = 1030 = 110-1 = .122.585 = 6

  • Evaluate without a calculatorlog381 = Log5125 =Log4256 =Log2(1/32) =3x = 815x = 1254x = 2562x = (1/32)434-5

  • Evaluating logarithms now you try some!Log 4 16 = Log 5 1 =Log 4 2 =Log 3 (-1) =(Think of the graph of y=3x) 20 (because 41/2 = 2) undefined

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

  • Natural logarithmslog e x = ln x

    ln means log base e

  • Common logarithmslog 10 x = log x

    Understood base 10 if nothing is there.

  • Common logs and natural logs with a calculatorlog10 button

    ln button

  • g(x) = log b x is the inverse off(x) = bx

    f(g(x)) = x and g(f(x)) = xExponential and log functions are inverses and undo each other

  • So: g(f(x)) = logbbx = x f(g(x)) = blogbx = x

    10log2 = Log39x =10logx =Log5125x = 2Log3(32)x =Log332x=2xx3x

  • Finding InversesFind the inverse of:y = log3xBy definition of logarithm, the inverse is y=3x OR write it in exponential form and switch the x & y! 3y = x 3x = y

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

  • Assignmentpg. 49048-55 all and 57-75 x 3s and 77-81 allThats only 20 problems.

  • Graphs of logsy = logb(x-h)+k Has vertical asymptote x=hThe domain is x>h, the range is all realsIf b>1, the graph moves up to the rightIf 0
  • Graph y = log1/3x-1Plot (1/3,0) & (3,-2)Vert line x=0 is asy.Connect the dots

    X=0

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

  • AssignmentPg. 951 50-63 all and finish yesterdays assignment