1. Properties of Logarithmic Functions
Section 5.4
Properties of Logarithmic Functions

2. Objectives
Convert from logarithms of products, powers, and quotients to expressions in terms of individual logarithms, and conversely.
Simplify expressions of the type logaax and

3. Logarithms of Products
The Product Rule For any positive numbers M and N and any logarithmic base a, loga MN = loga M + loga N. (The logarithm of a product is the sum of the logarithms of the factors.)

4. Example
Express as a single logarithm:

5. Logarithms of Powers
The Power Rule For any positive number M, any logarithmic base a, and any real number p, (The logarithm of a power of M is the exponent times the logarithm of M.)

6. Example
Express as a product.

7. Logarithms of Quotients
The Quotient Rule For any positive numbers M and N, and any logarithmic base a, (The logarithm of a quotient is the logarithm of the numerator minus the logarithm of the denominator.)

8. Example
Express as a difference of logarithms:

9. Example
Express as a single logarithm:

10. Example
Express each of the following in terms of sums and differences of logarithms.

11. Example (continued)

12. Example (continued)

13. Example
Express as a single logarithm:

14. Example
Given that loga 2 ≈ and loga 3 ≈ 0.477, find each of the following, if possible.

15. Examples (continued)
cannot be found using these properties and the given information.

16. Expressions of the Type loga ax
The Logarithm of a Base to a Power For any logarithmic base a and any real number x, loga ax = x. (The logarithm, base a, of a to a power is the power.)

17. Examples
Simplify.
a) loga a8 b) ln et c) log 103k
a. loga a8

18. Expressions of the Type
A Base to a Logarithmic Power For any base a and any positive real number x, (The number a raised to the power loga x is x.)

19. Example
Simplify.