Logarithms Logarithmic and Exponential Form

[Pages:4]Logarithms

A logarithm of a given number , is the exponent required for the base , to be raised to in order to produce that number .

log = = Note that means " "

Logarithmic and Exponential Form

Change logarithm equations to exponential form or exponential equations to logarithmic form using the definition of a logarithm.

Example: Given 43/2 = 8 , change the equation to logarithmic form.

Solution:

Compare the equation to the definition and rewrite it.

Definition: log = = Given: 43/2 = 8

Notice that = 4, = 8, and 3

= 2 , respectively.

Therefore, using the definition: 43/2 = 8

=

Example:

Given

log25

5

=

1 2

,

change

the

equation

to

exponential

form.

Solution:

Compare the equation to the definition and rewrite it.

Definition: log = = 1

Given: log25 5 = 2

Notice that = 25, = 5, and 1

= 2 , respectively.

Therefore,

using

the

definition:

log25

5

=

1 2

/ =

This instructional aid was prepared by the Tallahassee Community College Learning Commons.

Solving Logarithm and Exponential Equations

Evaluate logarithmic equations by using the definition of a logarithm to change the equation into a form that can then be solved.

Example: Given 3-1 = 7 , solve for .

Solution:

Step 1: Set up the equation and use the definition to change it.

Definition: log = = Given 3-1 = 7

Notice 3 is the base or , and 7 is the given number.

3-1 = 7 log3 7 = - 1

Step 2: Now use the properties of logarithms to solve.

Recall the Change of Base Property: log

log = log

Apply it to log3 7.

log 7 log3 7 = log 3

Step 3: Use the order of operations to finish solving for . log 7

- 1 = log 3

= +

Example: Given log6( + 2) = 3 , solve for .

Solution:

Step 1: Set up the equation and use the definition to change it.

Definition: log = = Given log6( + 2) = 3 Notice 6 is the base or , and 3 is the exponent or .

log6( + 2) = 3 63 = + 2

Step 2: Now use the order of operations to solve.

63 = + 2 216 = + 2 214 =

=

This instructional aid was prepared by the Tallahassee Community College Learning Commons.

Expanding and Simplifying Logarithms

To expand or simplify logarithms, utilize the various properties of logarithms in conjunction with the definition.

92

Example:

Given

log3

( 2

+

) , expand

1

the

logarithm.

Solution:

Step 1: Expand the expression using the properties of logarithms.

Step 2: Now simplify further using the properties of logarithms and the definition.

Recall the Logarithm Multiplication

and Division Properties:

log

=

log

+

log

log ( ) = log - log

Apply them to 92 and 2 + 1.

92

Given

log3

( 2

+

): 1

log3 9 + log3 2 - log3 (2 + 1)

Recall the Logarithm for Powers Property: log = log

Apply it to the 2and log3 (2 + 1).

log3 9 + log3 2 - log3 (2 + 1)

log3 9 + log3 2 - log3(2 + 1)1/2

log3

9

+

2

log3

-

1 2

log3(2

+

1)

By definition, log3 9 = 2 since 32 = 9,

so our final answer becomes:

+

-

(

+

)

Example: Write 3 log2 - log2 - 7 log2 as a single logarithm.

Solution:

To simplify the expression, work backwards with the logarithmic properties.

Step 1: Use the Logarithm for Powers Property where appropriate.

Given: 3 log2 - log2 - 7 log2 Notice that it can be applied to 3 log2 and 7 log2 .

3 log2 - log2 - 7 log2 log2 3 - log2 - log2 7

Step 2: Simplify using the Logarithm Multiplication and Division Properties. Use the order of operations as a guide.

log2 3 - log2 - log2 7 log2 3 - (log2 + log2 7) log2 3 - log2 7

This instructional aid was prepared by the Tallahassee Community College Learning Commons.

Solving Expanded Logarithms

Solving expanded logarithms requires applying the definition of logarithms and all the logarithm properties as needed.

Example: Given ln( - 2) + ln( - 3) = ln(2 + 24) , solve for .

Solution:

Note: ln( - 2) is only valid if 2, ln( - 3) is only valid if 3, and ln(2 + 24) is only valid if -12. For the equation to be valid, all conditions must be met, so 3.

Step 1: Simplify the left side of the equation using the multiplication and division properties of logarithms.

ln( - 2) + ln( - 3) = ln(2 + 24)

ln( - 2)( - 3) = ln(2 + 24) ln(2 - 5 + 6) = ln(2 + 24)

Step 2: Use logarithm properties. Recall logarithm properties of bases: ln = and ln =

ln(2 - 5 + 6) = ln(2 + 24) Let both sides of the equation become the exponent of the base , and apply the property.

ln(2-5+6) = ln(2+24) 2 - 5 + 6 = 2 + 24

Step 3: Combine like terms to solve for . 2 - 5 + 6 = 2 + 24 2 - 7 - 18 = 0 ( - 9)( + 2) = 0 = 9, -2

Step 4: Check your answers. Recall that every logarithm must meet the conditions for the answer to be correct.

For = 9 ln((9) - 2) + ln((9) - 3) = ln(2(9) + 24) ln(7) + ln(6) = ln(42) ln(7 6) = ln(42) This is valid!

For = -2 Since -2 3, it does not meet all the conditions, and is not valid.

Therefore: =

Practice Exercises:

1. Given log4(-) + log4(6 - ) = 2,

Solve for x.

2.

Expand

log2

(

2

) -1

completely.

3. Write the following as a single

logarithm: 2 log3 + 4 - 8 log3

Answers:

1. = -2

1

1

2. log2 - 2 log2( - 1) - 2 log2( + 1)

813 3. log3 8

This instructional aid was prepared by the Tallahassee Community College Learning Commons.

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