UNIT 11 LESSON 3

PROPERTIES OF LOGARITHMS

COLLEGE PREP

OBJECTIVES

• Understand properties of logarithms

• Expand and condense logarithms

• Evaluate logarithms without common bases (10,𝑒)

Inverse Property

INVERSE PROPERTY OF LOGARITHMS

𝑎log𝑎 𝑀 = 𝑀 log𝑎 𝑎

𝑟 = 𝑟

Logs and exponentials are inverses. If their bases match, they cancel each other out.

Inverse Property

Evaluate each logarithm.

A) 3log3 17 B) 0.5log0.5 11

C) log2 2 6 D) ln 𝑒7

Product Rule

PRODUCT RULE OF LOGARITHMS

log𝑎 𝑀𝑁 = log𝑎𝑀 + log𝑎 𝑁

Product Rule

Write each expression as the sum of logs.

E) log3(9𝑥) F) ln(2𝑘)

Quotient Rule

QUOTIENT RULE OF LOGARITHMS

log𝑎 𝑀

𝑁 = log𝑎𝑀 − log𝑎 𝑁

Quotient Rule

Write each expression as the difference of logs.

G) log4 𝑥

16 H) log

𝑠

𝑡

Expand Logarithms

Expand the logarithm completely.

I) log4 3𝑥

𝑦 J) ln

𝑎

5𝑏𝑐

Power Rule

POWER RULE OF LOGARITHMS

log𝑎𝑀 𝑟 = 𝑟𝑙𝑜𝑔𝑎𝑀

Power Rule

Expand the logarithm completely.

K) log7 2 4 L) ln 𝑥

Expand Logarithms

Expand the logarithm completely.

M) log3(9𝑥 2𝑦4) N)log

100𝑥

𝑦

Condense Logarithms

Condense the logarithm completely.

O) log4 2 + log4 8 P) ln 𝑥 + 3 − ln 𝑥

Condense Logarithms

Condense the logarithm completely.

Q) 1

2 log3 𝑥 + 2 + 2 log3 𝑥

Condense Logarithms

Condense the logarithm completely.

R) log2 9 + 2 log2 𝑥 − log2 𝑥 − 4

Change of Base Formula

To calculate logarithms that are not of a common base

with a calculator, use the CHANGE OF BASE FORMULA.

log𝑎𝑀 = log𝑀

log 𝑎 𝑜𝑟

ln𝑀

ln 𝑎

Change of Base Formula

Evaluate the logarithm to 3 decimal places.

S)log3 11 T)log2 9

U)log6 40 V) log51 1

Write as a single log:

CHALLENGE: Condense the logarithm completely

log 𝑥 + 2 log 𝑦 − 3 log 𝑧 + 1

2 log 5 − log 10 − 4 log𝑚

Expand:

CHALLENGE: Expand the logarithm completely

ln 5𝑥𝑦2𝑤

6𝑧 1 3𝑚𝑛

OBJECTIVES

• Understand properties of logarithms

• Expand and condense logarithms

• Evaluate logarithms without common bases (10,𝑒)

HOMEWORK

In Class Assignment C11.3

Homework H11.3

EDPuzzle Q11.3