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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.

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Logarithmic expressions and equations

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
Pre-requisite Check

Simplify

Answers: 5, 1, 4, 24


Objectives
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
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

ANSWER

1.366

3.

log5 49

ANSWER

2.418

4.

ANSWER

0.526

ANSWER

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

ANSWER

log2 5

+

5x

6.

log 2x 3

3 log x

ANSWER

log 2

+

log3

7

7.

ANSWER

8.

log6 4

2 log6x

log6y

ANSWER

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

ANSWER

log5 3

y

10.

log2 7

log2 5

ANSWER

+

log2 35

11.

log 4

2log 3

log 36

ANSWER

+

12.

3 log x

log y

log

ANSWER

Condense the expression. Assume all variables are positive.


Objective 2 solving equations
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
1. Equal Powers Property

Solve the equation




2 equal logarithms property
2. Equal Logarithms Property

Solve

  • Solution:


Exponentiate both sides
Exponentiate Both Sides

Solve

  • Solution:


Objective 3 change base formula
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 Base Formula

Change-of-base 6 for

Answer:

Change-of-base to the common logarithm of

Answer:


Conclusion
Conclusion

Summary

Assignment

  • How do you solve logarithmic equations?

    • If the equations can be rewritten so they have the same base and if a single variable appears as an exponent, take the logarithm of each side and solve for the variable; if the two sides of the equation can be written as logarithms to the same base, set the logarithms equal.

  • Pg445 #(22-33, 35-57 ODD)

  • Pg452 #(27-33, 37-51 ODD)

  • Mid-Unit Quiz Monday/Tuesday

  • UNIT Exam next week on Thursday/Friday.


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