3.7: Properties of Logarithms
What is the value of the expression ?
Alone, neither of these expressions has an integer value, therefore combining them might seem like a bit of a challenge. The value of log _{ 6 } 8 is between 1 and 2; the value of log _{ 6 } 27 is also between 1 and 2.
Is there an easier way?
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James Sousa: The Properties of Logarithms
Guidance
Previously we defined the logarithmic function as the inverse of an exponential function, and we evaluated log expressions in order to identify values of these functions. In this lesson we will work with more complicated log expressions. We will use the properties of logarithms to write a log expression as the sum or difference of several expressions, or to write several expressions as a single log expression.
Properties of Logarithms
Because a logarithm is an exponent, the properties of logs reflect the properties of exponents.
The basic properties are:
Expanding Expressions
Using the properties of logs, we can write a log expression as the sum or difference of simpler expressions. Consider the following examples:
 = =
 = =
Using the log properties in this way is often referred to as "expanding". In the first example, expanding the log allowed us to simplify, as log _{ 2 } 8 = 3. Similarly, in the second example, we simplified using the log properties, and the fact that log _{ 3 } 3 = 1.
Condensing Expressions (Answer to the concept question in the introduction)
To condense a log expression, we will use the same properties we used to expand expressions. Consider the expression . Individually, neither of these expressions has an integer value. The value of log _{ 6 } 8 is between 1 and 2; the value of log _{ 6 } 27 is also between 1 and 2.
However, if we condense the expression, we get:
Example A
Expand each expression:
a.  b. 

Solution:
 a.

b.

Example B
Condense the expression:
 2log _{ 3 } x + log _{ 3 } 5 x  log _{ 3 } ( x + 1)
Solution:
Note that not all solutions may be valid, since the argument must be defined. For example, the expression above: is undefined if x = 1.
Example C
Condense the expression:
log _{ 2 } (x ^{ 2 }  4)  log _{ 2 } ( x + 2)
Solution:
Note that the argument of a log must be positive. For example, the expressions in Example 'C' above are not defined for x ≤ 2 (which allows us to "cancel" ( x +2) without worrying about the condition x ≠ 2). >
Guided Practice
1) Condense the following expressions into a single logarithm:
2) Condense the expression into a single logarithm:
3) Condense the following into a single logarithm:
4) Expand the logarithm:
Answers
1) To condense the logs, apply the rule as explained in the lesson above:
2) Recall that
3) Recall that
4) Reversing the rule used in Q 2 gives:
 (reducing the fraction first)
Explore More
Expand each logarithmic expression:
 If expand
 If expand
Condense each logarithmic expression:
Simplify:
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Description
Learning Objectives
Here you will learn how to use the properties of logarithms to rewrite more complex logarithmic expressions as sums and differences of simpler expressions, and also to combine multiple expressions into a single logarithmic expression.