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Expanding Logarithmic Expressions (page 2 of 5)

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:

     

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      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 2002-2011 All Rights Reserved

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


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Cite this article as:

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

 



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