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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:




      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)

You can use the Mathway widget below to practice "Logarithmic Expressions and Equations", subtopic "Expanding Logarithmic Expressions". Try the entered exercise, or type in your own exercise. Then click "Answer" to compare your answer to Mathway's. (Or skip the widget and continue with the lesson.)

(Clicking on "View Steps" on the widget's answer screen will take you to the Mathway site, where you can register for a free seven-day trial of the software.)

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Cite this article as:

Stapel, Elizabeth. "Expanding Logarithmic Expressions." Purplemath. Available from Accessed


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