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Simplifying Logarithmic Expressions (page 3 of 5)

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

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). Copyright © Elizabeth Stapel 2002-2011 All Rights Reserved

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:

You can use the Mathway widget below to practice "Logarithmic Expressions and Equations", subtopic "Simplifying 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:

(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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