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Basic Log Rules / Expanding
     Logarithmic Expressions (page 1 of 4)

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

You have learned various rules for manipulating and simplifying expressions with exponents, such as the rule that says that x³ · x⁵ equals x⁸ because you can add the exponents. There are similar rules for logarithms.

Log Rules:

1) log_b(mn) = log_b(m) + log_b(n)

2) log_b(^m/_n) = log_b(m) – log_b(n)

3) log_b(mⁿ) = n · log_b(m)

In less formal terms, the log rules might be expressed as:

1) Multiplication inside the log can be turned into addition outside the log, and vice versa.

2) Division inside the log can be turned into subtraction outside the log, and vice versa.

3) An exponent on everything inside a log can be moved out front as a multiplier, and vice versa.

(Just as when you're dealing with exponents, the above rules work only if the bases are the same. For instance, the expression "log_d(m) + log_b(n)" cannot be simplified, because the bases are not the same, just as x² · y³ cannot be simplified.)

Expanding logarithms

Log rules can be used to simplify expressions or to solve for values. Here are some typical example problems:

• Expand log₃(2x).

When they say to "expand", they mean that they've given you one log expression with lots of stuff inside it, and they want you to use the log rules to take the log apart into lots of separate logs, each with only one thing inside. That is, they've given you one log with a complicated argument, and they want you to convert this to many logs, each with a simple argument. In this case, we have "2x" inside the log. Since this is multiplication, I can take it apart as addition outside the log, so:

log₃(2x) = log₃(2) + log₃(x)

The answer they are looking for is:

log₃(2) + log₃(x)

Do not try to evaluate "log₃(2)" in your calculator. While you would be correct that "log₃(2)" is just a number, for the answer to this question they're actually looking for the exact form of the log, as shown above, and not a decimal approximation.

• Expand log₄( ¹⁶/_x ).

I have division inside the log, which can be split apart as subtraction outside the log, so:

log₄( ¹⁶/_x ) = log₄(16) – log₄(x)

Note, however, that the first term in the answer can be simplified to an exact value. That is:

log₄(16) = 2

(You should always remember to take the time to check to see if anything can be simplified.)

Then the original expression expands as:

log₄( ¹⁶/_x ) = 2 – log₄(x)

• Expand log₅(x³).   Copyright © Elizabeth Stapel 2006-2008 All Rights Reserved

The exponent inside can be taken out front as a multiplier:

log₅(x³) = 3 · log₅(x) = 3log₅(x)

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