Logarithmic Expressions and Equations

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# Logarithmic Expressions and Equations - PowerPoint PPT Presentation

Logarithmic Expressions and Equations. 14.0 Students understand and use the properties of logarithms to simplify logarithmic numeric expressions and to identify their approximate values.

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### Logarithmic Expressions and Equations

14.0 Students understand and use the properties of logarithms to simplify logarithmic numeric expressions and to identify their approximate values.

11.1 Students understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.

11.2 Students judge the validity of an argument according to whether the properties of real numbers, exponents, and logarithms a have been applied correctly at each step.

Pre-requisite Check

Simplify

Objectives
• Use properties of logarithms to evaluate, expand, or condense such expressions to be able to solve problems involving logarithms.
• Solve exponential and logarithmic equations.
• Rewrite equivalent logarithm expression by changing base (if we have time, else next week)
Objective 1: Logarithm Properties
• Let b, m, and n be positive numbers such that b≠1.
• Product Property
• Quotient Property
• Power Property

Example 1

Use log7 2 0.356 and log7 5 0.827 to find the value of the expression to the nearest thousandth.

2

2

a.

b.

c.

log7

log7

10

log7

25

5

5

SOLUTION

a.

log7

Quotient property

log7 2 log7 5

=

0.356 0.827

Use the given values of log7 2 and log7 5.

0.471

=

Simplify.

Use Properties of Logarithms

Example 1

=

(

)

Product property

log7 2 log7 5

2

5

+

=

b.

log7

10

Use the given values of log7 2 and log7 5.

0.356 0.827

+

1.183

=

Simplify.

c.

log7

25

log7 52

Express 25 as a power.

=

2 log7 5

Power property

=

Use the given value of log7 5.

(

)

0.827

2

Simplify.

1.654

=

Use Properties of Logarithms

log7

Express 10 as a product.

Example 2

3x

3x

a.

b.

b.

log4 5x2

log7

log7

y

y

SOLUTION

a.

log4 5x2

log45 log4x2

Product property

+

=

Power property

log452 log4x

+

=

log73x log7y

Quotient property

=

Expand a Logarithmic Expression

Expand the expression. Assume all variables are positive.

Example 2

Expand a Logarithmic Expression

log73 log7xlog7y

+

Product property

=

Example 3

a.

b.

log 16 2 log 2

3 log 5 log 4

+

SOLUTION

a.

log 16 2 log 2

log 16 log 22

Power property

=

Quotient property

=

Simplify.

log 4

=

16

log

22

Condense a Logarithmic Expression

Condense the expression.

Example 3

b.

3 log 5 log 4

log 53 log 4

+

Power property

+

=

Product property

log

(

)

53

4

=

Simplify.

log 500

=

Condense a Logarithmic Expression

Expand and Condense Logarithmic Expressions

Checkpoint

3

1.

log5 21

log5

7

2.

log5 9

1.366

3.

log5 49

2.418

4.

0.526

1.892

Use log5 3 0.683 and log5 7 1.209 to find the value of the expression to the nearest thousandth. (Calculator!!)

Expand and Condense Logarithmic Expressions

Checkpoint

5.

log2 5x

log2 x

log2 5

+

5x

6.

log 2x 3

3 log x

log 2

+

log3

7

7.

8.

log6 4

2 log6x

log6y

4x2

+

log6

y

log3 5

log3x

log3 7

+

Expand the expression. Assume all variables are positive.

Expand and Condense Logarithmic Expressions

Checkpoint

x3

9.

log5 12

log5 4

log5 3

y

10.

log2 7

log2 5

+

log2 35

11.

log 4

2log 3

log 36

+

12.

3 log x

log y

log

Condense the expression. Assume all variables are positive.

Objective 2: Solving Equations

1. Equal Powers

Property

2. Equal Logarithms Property

• For b>0 and b≠1,

IFF x=y

• Example:

If , then x=5

• For positive numbers b, x, and y where b≠1,

IFF x=y

• Example:
1. Equal Powers Property

Solve the equation

Objective 3: Change Base Formula
• Change-of-Base Formula
• Let x, b, and c be positive numbers such that b≠1 and c≠1. Then,
Change Base Formula

Change-of-base 6 for

Change-of-base to the common logarithm of