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?

### Properties of Logarithms

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.

### Examples

#### Example 1

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 2

Expand each expression.

#### Example 3

Condense the expression 2log_{3} *x* + log_{3} 5*x* - log_{3} (*x* + 1).

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 4

Condense the expression log_{2} (x^{2} - 4) - log_{2} (*x* + 2).

Note that the argument of a log must be positive. For example, the expressions above are not defined for *x* ≤ 2 (which allows us to "cancel" (*x*+2) without worrying about the condition *x*≠ -2).

#### Example 5

Condense the following into a single logarithm: .

Recall that

#### Example 6

Expand the logarithm: .

Reversing a previously used rule gives .

(reducing the fraction first)

### Review

Expand each logarithmic expression:

- If expand
- If expand

Condense each logarithmic expression:

Simplify:

### Review (Answers)

To see the Review answers, open this PDF file and look for section 3.7.