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Transformations of Logarithms

  1. Jul 23, 2004 #1
    We have just recently been doing transformations of sin and cos graphs, but we must find out transformations of logarithm graphs.

    A typical log function could be log (x).

    What i want to know is, when you change the base, what would happen to the graph, when you put a number out the front, what would happen to the graph, adding or subtracting a number in the brackets [ log (x +2) ] or even adding or subracting numbers at the end.

    Any help on what these situations would do to these graphs would be greatly appreciated, also, i have a situation where a log function cuts the y axis at around 2.1, and the x axis at around 1.7. We gotta use transformations to obtain the graph.

    So if any help can be provided to help make a log function that describes the situation above, or with transformations of logs in general, it would be much helpful. Thanks
     
  2. jcsd
  3. Jul 23, 2004 #2
    First lets see the base change:

    [tex]
    y = log_{a}x = \frac{log_{b}x}{log_{b}a}
    [/tex]

    If you replace b by a convenient number, say 10 or e, you will see that the denominator [tex]log_{b}a[/tex] is just a scaling factor and all you have to do is draw the graph of the numerator as such and fix the numbers on the axes according to the scaling factor.

    I'll give you an idea:

    [tex]
    y = f(x)
    [/tex]

    and

    [tex]
    y = f(x+a)
    [/tex]

    are two different graphs representing the same general shape though and to get the second graph from the first, shift the first graph left (along the x-axis)by a units if a is positive and to the right by -a units if a is negative. (Do you see why? If not, draw any random graph and experiment by taking general values of a)

    Also,

    [tex]
    y = y_{0} + f(x)
    [/tex]

    and

    [tex]
    y = f(x)
    [/tex]

    differ in that the first one has an intercept of [tex]\pm \|y_{0}\|[/tex] depending on the sign of [tex]y_{0}[/tex] and the second one passes through the origin (I am assuming of course that f(x) has no constant term...this is not an incorrect assumption since I can isolate a constant term from a function g(x) and write it as g(x) = f(x) + constant term).

    Now, think about the log(x+a) graphs yourself and we'll help you out if you have a problem still...

    ---------
    For the second problem, you can try a variety of logarithmic functions. The most general one could be

    [tex]
    y = y_{0} + y_{m}log_{b}(x+a)
    [/tex]

    Now adjust this according to the data given in your problem and solve it.
    ---------

    Hope that helps...

    Cheers
    Vivek
     
    Last edited: Jul 23, 2004
  4. Jul 23, 2004 #3
    [tex]\log{kx+a} + b[/tex]

    here's my way:
    -a = vertical asymptote
    b + 1 = horizontal asymptote
    k = amplitude
     
  5. Jul 23, 2004 #4

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    That's not a very good way.

    First, I assume you mean y= log(kx+ a)+ b since just (a+b) doesn't add anything new.

    Second, when x= -a, you have y= log(a- ka)+ b which may or may not exist but vertical is not a vertical asymptote unless k= 1. y= log(x) has a vertical asymptote at x= 0 so y= log(kx+ a)+ b has a vertical asymptote where kx+ a= 0 or x= -a/k.

    Third, y= log(kx+a)+ b does not have a horizontal aymptote for any values of k,a, or b.

    Fourth, since k is multiplies x, not y, it stretches the graph along the x-axis, not the y. In any case, since the range of log(kx+ a)+ b is all real numbers, it doesn't have an amplitude.
     
  6. Jul 24, 2004 #5
    woah, now im gettin really confused. As i stated above, my problem involved using transformations from log (x) to get a function that will look like one i have on my paper.

    I cant really draw it, the only thing is i know its a log function, and it would have to be that log (-x) thingy, cus its in that shape way. The only thing i could prolly say is it cuts the y-axis at 0.21, and has the other points (0.5, 0.19), (1, 0.15), (1.5, 0.1), (1.75, 0).

    I have consulted my teacher about it, and he said using the graph we have, use tranformations to form that function (ie educated guessing of numbers). Now obviously i gotta know what a number in what place does to the log function, but that is the problem.

    P.S. Thankyou so far for all your help guys.
     
    Last edited: Jul 24, 2004
  7. Jul 25, 2004 #6
    Well as I said you need to adapt a function to suit your needs according to a given situation. There is no surefire tip that we can give you that will work for all problems (that would make math very boring). You cannot know everything about graphing without making graphs yourself. Start with a simple log x graph. Multiply the argument by a constant, add or substract to or from the independent variable or multiply the function by a scalar...compressing or expanding the graph as the case may be. The more graphs you draw the better you become at analyzing these situations. Label your graphs well. Its always better to start from a principal (simple) function and then apply necessary transformations to it to get the graph for the final function.
     
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