1.Warm-Up 4/22

A 𝑥=3 𝑦−5

2

Rigor:You will learn how to evaluate, graph and use the

properties of logarithmic functions.

Relevance:You will be able to solve pH chemistry problems

using logarithmic functions.

3-2 Logarithmic Functions

Example 1: Evaluate each logarithm.

a. b. log 3 81=𝑥

3𝑥¿ 813𝑥¿ 34

𝑥¿ 4

log 5 √5=𝑥5𝑥¿ √55𝑥¿5

12

𝑥¿12

log 3 81=4 log 5 √5=12

c. d. log 7

149

=𝑥

7𝑥¿149

7𝑥¿7−2

log 7149

=−2

log 22=𝑥2𝑥¿2

log 22=1

Example 2: Evaluate each expression.

a. b. ¿ log553

¿3¿ 4.7

Example 3: Evaluate each expression.

a. b. ¿ log10−3

¿−3¿5

c. d. ≈1.4149 undefinedU se acalculator .

Example 4: Evaluate each expression.

a. b. ¿0.73

¿1.3862c. d. ¿6

undefined

U se acalculator .

Example 5a: Sketch and analyze the graph of the function. Describe its domain, range, intercepts, asymptotes, end behavior, and where the function is increasing, or decreasing.

𝑥=log 3 𝑦3𝑥=𝑦

x

– 3 .037

– 2 .111

– 1 .333

0 1

1 3

2 9

3 27

𝑓 (𝑥 )=log3 𝑥x y

x

.037 – 3

.111 – 2

.333 – 1

1 0

3 1

9 2

27 3

𝑓 −1 (𝑥 )=3𝑥

𝑓 (𝑥 )=log3 𝑥

Domain:Range:x-intercept:Asymptotes:End Behavior:

Increasing/Decreasing:

(0 ,∞)(−∞ ,∞)

(1 ,0)𝑥=0

lim𝑥→0+¿ 𝑓 ( 𝑥)=−∞¿

¿ lim𝑥→∞

𝑓 (𝑥)=∞and

I ncreasing :(0 ,∞)

Example 5b: Sketch and analyze the graph of the function. Describe its domain, range, intercepts, asymptotes, end behavior, and where the function is increasing, or decreasing.

𝑥=log 12

𝑦12

𝑥

=𝑦

x

– 3 8

– 2 4

– 1 2

0 1

1 .5

2 .25

3 .125

𝑔 (𝑥 )= log 12

𝑥

x y

x

8 – 3

4 – 2

2 – 1

1 0

.5 1

.25 2

.125 3

𝑔−1 (𝑥 )=12

𝑥

𝑔 (𝑥 )=log 12

𝑥

Domain:Range:x-intercept:Asymptotes:End Behavior:

Increasing/Decreasing:

(0 ,∞)(−∞ ,∞)

(1 ,0)𝑥=0

lim𝑥→0+¿ 𝑓 ( 𝑥)=∞ ¿

¿ lim𝑥→∞

𝑓 (𝑥)=−∞and

Decreasing :(0 ,∞)

Example 6a: Use the graph of to describe the transformation of the function. Then sketch both functions.

𝑘 (𝑥 )=log (𝑥+4 )

x y

– 4 V.A.

– 3 0

– 2 .30103

– 1 .47712

0 .60206

1 .69897

2 .77815

𝑓 (𝑥 )=log 𝑥

x y

0 V.A.

1 0

2 .30103

3 .47712

4 .60206

5 .69897

6 .77815

𝑘 (𝑥 )= 𝑓 (𝑥+4)The graph of is the graph of translated 4 units to the left.

Example 6b: Use the graph of to describe the transformation of the function. Then sketch both functions.

𝑚 (𝑥 )=− log𝑥−5

x y

0 V.A.

1 – 5

2 – 5.3010

3 – 5.4771

4 – 5.6020

5 – 5.6989

6 – 5.7781

𝑓 (𝑥 )=log 𝑥

x y

0 V.A.

1 0

2 .30103

3 .47712

4 .60206

5 .69897

6 .77815

𝑚 (𝑥 )=− 𝑓 (𝑥 )−5The graph of is the graph of is reflected in the x-axis and translated 5 units down.

Example 6c: Use the graph of to describe the transformation of the function. Then sketch both functions.

𝑝 (𝑥 )=3 log (𝑥+2 )

x y

– 2 V.A.

– 1 0

0 .90309

1 1.4314

2 1.8062

3 2.0969

4 2.3345

𝑓 (𝑥 )=log 𝑥

x y

0 V.A.

1 0

2 .30103

3 .47712

4 .60206

5 .69897

6 .77815

𝑝 (𝑥 )=3 𝑓 (𝑥+2)The graph of is the graph of is expanded vertically by a factor of 3 andtranslated 2 units to the left.

√−1math!

3-2 Assignment: TX p178, 4-40 EOE

Test Corrections Due Friday 4/25Chapter 3 test Thursday 5/1