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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:
     
            

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:

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 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:

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