1 of 21 Pre-Calculus Chapter 3.3 – 3.4 Warm - up.

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Pre-Calculus Chapter 3.3 – 3.4

Warm - up

3

45

2

11610

4

144

3

49

18 .12

25 .11

53 .10

08 .9

xz

yx

ba

completelySimplify

3 102 .9

15

1 .10

655 .11 ba

2

22

7

23 .12

z

zyx

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Chapter 3 Sec 3

Properties of Logarithms

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Essential Question

How do you rewrite logarithmic expressions to simplify or

evaluate them?

Key Vocabulary:Change-of-base formula

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Change of Base Formula• This allows you to write equivalent logarithmic

expressions that have different bases. For example change base 3 into base 10

a

xx

b

b

ba log

loglog

Base

a

xx

e

a ln

lnlog

Base

a

xxa

10

10

log

loglog

10 Base

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Change of BaseChange bases using common logarithms. Then approximate its value.

4log

25log25log a.

10

104

3219.26021.

3980.1

3log

18log18log b.

10

103

6309.24771.

2553.1

Change bases using natural logarithms. Then approximate its value.

4ln

25ln25log c. 4

3219.238629.1

21888.3

2ln

12ln12log d. 2

585.3693147.

48491.2

Example 1

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Properties of LogarithmsLet a be a positive number such that a ≠ 1, and let n be a real number. If u and v are positive real numbers, the following properties are true.

1. Product Property:

2. Quotient Property:

3. Power Property:

vuvuuv aaa lnlnuvln logloglog

vuv

uvu

v

uaaa lnlnln logloglog

unuunu na

na lnln loglog

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Example 2Write each logarithm in terms of ln 2 and ln 3.

6ln a. 32ln 27

2ln b. 27ln2ln

5

17log c. 5

7

3ln2ln 33ln2ln 3ln32ln

Use properties of logarithms to verify:

5

1

75

7 7log7log 7log5

17

15

11

7log

7log7log 7

5

1

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Example 3Expand each logarithmic expression.

yx 23log a.8

14ln b.

x

yx loglog3log 2 8ln14ln 2

1

x

yx loglog23log 8ln14ln2

1 x

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Example 4Condense each logarithmic expression.

3log5log3

1 a. xx xx ln24ln4 b.

2loglog5

1 c. 33 xx

53 3loglog xx 24 ln4ln xx

2

44ln

x

x

53 3log xx

2log5

1 3 xx

53 2log xx

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Essential Question

How do you rewrite logarithmic expressions to simplify or evaluate them?

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Chapter 3 Sec 4

Solving Exponential and Logarithmic Equations

Essential Question

How do you solve exponential and logarithmic equations?

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Solving Equations• One-to-One

ax = ay if and only if x = yloga x = loga y if and only if x = y

• Inverse Propertiesxa xa log

xa xa log

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a. 2x = 32

b. ln x – ln 3 = 0

c.

d. ex = 7

e. ln x = –3

f. log x = –1

One-to-One

One-to-One

One-to-One

Inverse

Inverse

Inverse

93

1

x

Example 1

Original Equation

RewrittenEquation Solution

2x = 25

ln x = ln 3

3–x = 32

ln ex = ln 7

eln x = e–3

10log x = 10–1

x = 5

x = 3

x = –2

x = 7

x = e–3

x = 10–1 = 0.1

Property

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Solve each equation.

a. ex = 72ln ex = ln 72x = ln 72 ~ 4.28

Example 2

b. 3(2x) = 42 2x = 14 log22x = log214x = log21481.3

2ln

14lnx

234 c. 2 xe54 2 xe

4

52 xe4

5lnln 2 xe

4

5ln2 x 11.0

4

5ln

2

1x

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Solve the equation.Example 3

11432 52 t

1532 52 t

2

153 52 t

2

15log3log 3

523 t

5.7log52 3t

5.7log52 3t

42.35.7log2

1

2

53 t

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Solve the equation using quadratics.Example 4

0232 xx ee

0232

xx ee

012 xx ee

02 xe 01 xe

2xe 1xe1lnx

0x

69.02ln x

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a. ln 3x = 2eln 3x = e2 3x = e2

Example 5

4ln25 c. x

46.23

1 2 ex

7log15log b. 33 xx

715 xx

2

84

x

x

1ln2 x

2

1ln x

2

1ln

ee x 61.02

1

ex

43log2 d. 5 x

23log 5 x23log 55 5 x

3

25

253

x

x

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Solve the equation check for extraneous solutions.

Example 6

xxx ln232ln2ln

2ln322ln xxx

22 ln672ln xxx 22 672 xxx

0672 xx

016 xx

101

606

xx

xx

Check:

36ln36ln

6ln29ln4ln

6ln2312ln26ln

6

x

1ln21ln1ln

1ln232ln21ln

1

x

ln (–1) is invalid.

X

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You have deposited $500 in an account that pays 6.75% interest, compounded continuously. How long will it take your money to double?

Example 5

te 0675.5001000 tr t ePeA 0 675.500te 0675.2

te 0675.ln2ln t0675.2ln

27.100675.

2lnt

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Essential Question

How do you solve exponential and

logarithmic equations?

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Daily Assignment• Chapter 3.3 -3.4• Text Book

• Pgs 211 – 212 • #1 – 45 and 59 – 73 Mode 4 (1,5,9,13…73)

• Pgs 221 – 222 • #1 – 21 Mode 4; #29 – 49 Mode 4;

#85 – 97 Mode 4• Read Section 3.5

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Ch 3aPop Quiz

3 Find

123For .3

9

3Simplify .2

276624

GCF) find :(hint completelyFactor .1

2

322

2

23

xf

xxxf

zyx

xyz

xxx

1 of 21 Pre-Calculus Chapter 3.3 – 3.4 Warm - up.

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