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 = 𝑀log𝑎 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 26 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

log𝑎𝑀

𝑁= log𝑎𝑀 − log𝑎 𝑁

Quotient Rule

Write each expression as the difference of logs.

G) log4𝑥

16H) log

𝑠

𝑡

Expand Logarithms

Expand the logarithm completely.

I) log43𝑥

𝑦J) ln

𝑎

5𝑏𝑐

Power Rule

POWER RULE OF LOGARITHMS

log𝑎𝑀𝑟 = 𝑟𝑙𝑜𝑔𝑎𝑀

Power Rule

Expand the logarithm completely.

K) log7 24 L) ln 𝑥

Expand Logarithms

Expand the logarithm completely.

M) log3(9𝑥2𝑦4) 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

2log3 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 𝑎𝑜𝑟

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

2log 5 − log 10 − 4 log𝑚

Expand:

CHALLENGE: Expand the logarithm completely

ln5𝑥𝑦2𝑤

6𝑧13𝑚𝑛

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