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Expanding and Simplifying
     Logarithmic Expressions
(page 2 of 4)

Sections: Basic log rules, Expanding & simplifying, Trick questions, Change-of-Base formula


  • Expand the following:
       
              log_2 (8x^4 / 5)

    The 5 is divided into the 8x4, so split the numerator and denominator by using subtraction:

      log_2 (8x^4 / 5) = log_2 (8x^4) - log_2 (5)

    Don't take the exponent out front yet; it is only on the x, not the 8, and you can only take the exponent out front if it is "on" everything inside the log. The 8 is multiplied onto the x4, so split the factors by using addition:

      log2(8x4) – log2(5) = log2(8) + log2(x4) – log2(5)

    The x has an exponent (which is now "on" everything inside its log), so move the exponent out front as a multiplier:

      log2(8) + log2(x4) – log2(5)

          = log2(8) + 4log2(x) – log2(5)

    Since 8 is a power of 2, I can simplify the first log to an exact value:

      log2(8) + 4log2(x) – log2(5)

         = 3 + 4log2(x) – log2(5)

    Each log contains only one thing, so this is fully simplified. The answer is:

      3 + 4log2(x) – log2(5)

  • Expand the following:
      
              log_3 [ (4(x - 5)^2) / (x^4(x - 1)^3) ]

    Use the log rules, and don't try to do too much in one step:

log_3(4) + 2log_3(x - 5) - 4log_3(x) - 3log_3(x - 1)

    Then the final answer is:   Copyright © Elizabeth Stapel 1999-2009 All Rights Reserved

      log3(4) + 2log3(x – 5) – 4log3(x) – 3log3(x – 1)


Simplifying logarithms

The logs rules work "backwards", so you can simplify ("compress"?) log expressions. When they tell you to "simplify" a log expression, this usually means they will have given you lots of log terms, each containing a simple argument, and they want you to combine everything into one log with a complicated argument. "Simplifying" in this context usually means the opposite of "expanding".

  • Simplify log2(x) + log2(y).

    Since these logs have the same base, the addition outside can be turned into multiplication inside:

      log2(x) + log2(y) = log2(xy)

    The answer is log2(xy).

  • Simplify log3(4) log3(5).

    Since these logs have the same base, the subtraction outside can be turned into division inside:

      log3(4) log3(5) = log3(4/5)

    The answer is log3(4/5).

  • Simplify 2log3(x).

    The multiplier out front can be taken inside as an exponent:

      2log3(x) = log3(x2)

  • Simplify 3log2(x) – 4log2(x + 3) + log2(y).

    I will get rid of the multipliers by moving them inside as powers:

      3log2(x) – 4log2(x + 3) + log2(y)
           = log2(x3) – log2((x + 3)4) + log2(y)

    Then I'll put the added terms together, and convert the addition to multiplication:

      log2(x3) – log2((x + 3)4) + log2(y)
           = log2(x3) + log2(y)log2((x + 3)4)
           = log2(x3y) – log2((x + 3)4)

    Then I'll account for the subtracted term by combining it inside with division:

      log_2(x^3y) – log_2((x + 3)^4) = log_2[ x^3y / (x + 3)^4 ]

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Cite this article as:

Stapel, Elizabeth. "Expanding and Simplifying Logarithmic Expressions." Purplemath. Available from
    http://www.purplemath.com/modules/logrules2.htm. Accessed
 

 



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