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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:
     
            

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

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:
    
            

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

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.)

To activate, click in the "Enter Problem" box below:

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