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:

logb(xy)=logbx+logby logb(xy)=logbx−logby logbxn=nlogbx

#### 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:

log28x =log28+log2x =3+log2x log3(x23) =log3x2−log33 =2log3x−1

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 _{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.

log525x2y

log10(100x9b)

#### 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: *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

### Review

Expand each logarithmic expression:

log5(ab) log6a3√b log6abc - If
v=logx(4z2y3) expandv log2(4x3y√) - If
R=log3(2GMc2) expandR

Condense each logarithmic expression:

log5A+log5C 12log2C−log2B 2logbx+2logby 6log10a+log10b 2log3a+4log3b−log3c - \begin{align*}\frac{1}{2}log_4 w - 5log_4 z\end{align*}
- \begin{align*}(log_{10} x + log_{10} y) - log_{10} w\end{align*}

Simplify:

- \begin{align*}log_{10} A^3 - log_{10} B^{\frac{2}{3}} + log_{10} A^{\frac{1}{3}} + log_{10} B^{\frac{5}{3}}\end{align*}
- \begin{align*}\frac{log_{9} A^2 - 2log_{9} B}{log_{9} A^2 + log_{9} B^3}\end{align*}
- \begin{align*}2ln(AB) - ln(\frac{B}{A})\end{align*}

### Review (Answers)

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