LOGARITHMS Section 4.2

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LOGARITHMS Section 4.2. JMerrill, 2005 Revised 2008. Exponential Functions. 1. Graph the exponential equation f(x) = 2 x on the graph and record some ordered pairs. Review. 2. Is this a function? Yes, it passes the vertical line test (which means that no x’s are repeated) 3. Domain? - PowerPoint PPT Presentation

Transcript of LOGARITHMS Section 4.2

  • LOGARITHMSSection 4.2JMerrill, 2005Revised 2008

  • Exponential Functions1. Graph the exponential equation f(x) = 2x on the graph and record some ordered pairs.

    xf(x)01122438

  • Review2. Is this a function?Yes, it passes the vertical line test (which means that no xs are repeated)

    3. Domain?

    Range?

  • Review2. Is the function one-to-one? Does it have an inverse that is a function?Yes, it passes the horizontal line test.

  • InversesTo graph an inverse, simply switch the xs and ys (remember???)

    f(x) = f -1(x) =

    xf(x)01122438

    xf(x)10214283

  • Now graphf(x)

    f-1(x)

  • How are the Domain and Range of f(x) and f -1(x) related? The domain of the original function is the same as the range of the new function and vice versa.

    f(x) = f -1(x) =

    xf(x)01122438

    xf(x)10214283

  • Graphing Both on the Same GraphCan you tell that thefunctions are inversesof each other? How?

  • Graphing Both on the Same GraphCan you tell that thefunctions are inversesof each other? How?

    They are symmetricabout the line y = x!

  • Logarithms and ExponentialsThe inverse function of the exponential function with base b is called the logarithmic function with base b.

  • Definition of the Logarithmic FunctionFor x > 0, and b > 0, b 1y = logbx iff by = x

    The equation y = logbx and by = x are different ways of expressing the same thing. The first equation is the logarithmic form; the second is the exponential form.

  • Location of Base and Exponent

    Logarithmic: logbx = y

    Exponential: by = xExponentBaseExponentBaseThe 1st to the last = the middle

  • Changing from Logarithmic to Exponential Forma.log5 x = 2 means52 = xSo, x = 25b.logb64 = 3meansb3 = 64So, b = 4 since 43 = 64You do:c. log216 = xmeansSo, x = 4 since 24 = 16d. log255 = x meansSo, x = since the square root of 25 = 5!2x = 1625x = 5

  • Changing from Exponential to Logarithmica.122 = xmeans log12x = 2b.b3 = 9means logb9 = 3

    You do:c. c4 = 16meansd. 72 = xmeanslogc16 = 4log7x = 2

  • Properties of LogarithmsBasic Logarithmic Properties Involving One:logbb = 1 because b1 = b.logb1 = 0 because b0 = 1

    Inverse Properties of Logarithms:logbbx = x because bx = bxblogbx = x because b raised to the log of some number x (with the same base) equals that number

  • Characteristics of GraphsThe x-intercept is (1,0). There is no y-intercept.The y-axis is a vertical asymptote; x = 0.Given logb(x), If b > 1, the function is increasing. If 0
  • TransformationsVertical Shift Vertical shiftsMoves the same as all other functions!Added or subtracted from the whole function at the end (or beginning)

  • TransformationsHorizontal ShiftHorizontal shiftsMoves the same as all other functions!Must be hooked on to the x value!

  • TransformationsReflectionsg(x)= - logbxReflects about the x-axisg(x) = logb(-x)Reflects about the y-axis

  • TransformationsVertical Stretching and Shrinkingf(x)=c logbxStretches the graph if the c > 1Shrinks the graph if 0 < c < 1

  • TransformationsHorizontal Stretching and Shrinkingf(x)=logb(cx)Shrinks the graph if the c > 1Stretches the graph if 0 < c < 1

  • DomainBecause a logarithmic function reverses the domain and range of the exponential function, the domain of a logarithmic function is the set of all positive real numbers unless a horizontal shift is involved.

  • Domain Cont.Domain

    Domain

    Domain

  • Properties of Commons Logs

    General PropertiesCommon Logarithms(base 10)logb1 = 0 log 1 = 0 logbb = 1 log 10 = 1 logbbx = x log 10x = x blogbx = x 10logx = x

  • Properties of Natural Logarithms

    General PropertiesNatural Logarithms(base e)logb1 = 0 ln 1 = 0 logbb = 1 ln e = 1 logbbx = x ln ex = x blogbx = x elnx = x