CHAPTER 4 4-4 Properties of logarithms

Objectives Use properties to simplify logarithmic expressions.

Translate between logarithms in any base.

Definition of logarithms ◦ The logarithmic function is the function , . where b is any number such

that                   is equivalent to    The function is read "log base b of x".    

◦ From the definition above, you can see that every logarithmic equation can be written in an equivalent exponential form and vice versa.

◦ Because logarithms are exponents, you can derive the properties of logarithms from the properties of exponents

Properties of logarithms

Remember that to multiply powers with the same base, you add exponents.

Example 1◦ Express log

64 + log

69 as a single logarithm. Simplify.

Example 2◦ Express as a single logarithm. Simplify, if possible.

◦ A) log5625 + log

525

◦ B) log 1/3

27 + log191/3

Student Guided practice◦Do problems 1 -3 in your book page 260

Properties of logarithms

Remember that to divide powers with the same base, you subtract exponents

Example 3◦ Express log

5100 – log54 as a single logarithm. Simplify, if possible.

Example 4◦ Express log

749 – log7 7 as a single logarithm. Simplify, if possible.

Student guided practice◦Do problems 4 -6 in your book page 260

Properties of logarithms◦ Because you can multiply logarithms, you can also take powers of logarithms.

Example 5 ◦ Express as a product. Simplify, if possible.

◦A. log2326

◦B. log8420

◦ c. log2 ( 1/2 )5

Student guided practice◦Do problems 7-10 in your book page 260

Properties of logarithms ◦ Exponential and logarithmic operations undo each other since they are inverse

operations.

Example 6 ◦Simplify each expression.

◦ a. log3311

◦b. log381

◦ c. 5log510

Example 7◦Simplify 2log

2(8x)

Student guided practice◦Do problems 12-14 in your book page 260

Change of base formula◦Most calculators calculate logarithms only in base 10 or base e .You can change a

logarithm in one base to a logarithm in another base with the following formula.

Example 8◦ Evaluate log

328.

Example 9◦ Evaluate log

927.

Student guided practice◦Do problems 15-18 in your book page 260

Properties ◦ Logarithmic scales are useful for measuring quantities that have a very wide range

of values, such as the intensity (loudness) of a sound or the energy released by an earthquake.

◦ The Richter scale is logarithmic, so an increase of 1 corresponds to a release of 10 times as much energy.

Geology Application

◦ The tsunami that devastated parts of Asia in December 2004 was spawned by an earthquake with magnitude 9.3 How many times as much energy did this earthquake release compared to the 6.9-magnitude earthquake that struck San Francisco in1989?

◦ Solution:

◦ The Richter magnitude of an earthquake, M, is related to the energy released in ergs E given by the formula.

Application#2◦How many times as much energy is released by an earthquake with

magnitude of 9.2 by an earthquake with a magnitude of 8?

Homework ◦Do odd problems from 20-35 in your book page 260

Have a great day!!!