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Help Understanding What a Logarithm IS

  1. Jun 22, 2006 #1
    I need help defining a logarithm.

    My book simply says: A logarithm is an exponent.

    This stumped me because I can't see how that is. I don't know what question to ask, but I might not be apprehending the relationship between an expo. function and a log. function.
     
  2. jcsd
  3. Jun 22, 2006 #2
    Well the logarithm base b is defined as

    y = logb(x)

    if

    x = by

    Notice that in the second equation y is the exponent, and in the first equation it is the logarithm, thus since these two equations are identical and express the exact same mathematical relationship it follows that a logarithm is essentially an exponent.
     
  4. Jun 22, 2006 #3
    Thanks, that helped a lot.

    I'm curious about why 'x' ended up where it is when the logarithm was converted to an exponential function or vice versa.
     
  5. Jun 22, 2006 #4
    When converting from an exponential to a log (We'll use x=by as an example), you bring the y down. This bumps the b down into the base (Making it x=by). Then you switch the x and the y (Finishing the conversion to y=logbx). Did that make sense?
     
  6. Jun 22, 2006 #5

    HallsofIvy

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    Science Advisor

    In technical terms, the logarithm is the "inverse" of the exponential function. That is, you swap the "x" and "y" values: if y= f(x) then x= f-1(y). As Ateowa said, if y= logax then x= ay.
     
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