Algebra and Trigonometry, 5th Edition

5.4 Properties of Logarithmic Functions

• Convert from logarithms of products, powers, and quotients to expressions in terms of individual logarithms, and conversely.

• Simplify expressions of the type ${\log }_a{a}^x$ and $a^{{\log }_{a}x}$.

We now establish some properties of logarithmic functions. These properties are based on the corresponding rules for exponents.

Logarithms of Products

JUST IN TIME 7

The first property of logarithms corresponds to the product rule for exponents: $a^m·a^n=a^{m+n}$.

The Product Rule

For any positive numbers M and N and any logarithmic base a,

$${\log }_{a}MN={\log }_{a}M+{\log }_{a}N.$$

(The logarithm of a product is the sum of the logarithms of the factors.)

Example 1

Express as a sum of logarithms: ${\log }_3(9·27)$.

Solution

We have

$$\begin{array}{l}{\log }_3(9·27)={\log }_{3}9\end{array}$$