<< Chapter < Page Chapter >> Page >

Given a logarithm with the form log b M , use the change-of-base formula to rewrite it as a quotient of logs with any positive base n , where n 1.

  1. Determine the new base n , remembering that the common log, log ( x ) , has base 10, and the natural log, ln ( x ) , has base e .
  2. Rewrite the log as a quotient using the change-of-base formula
    • The numerator of the quotient will be a logarithm with base n and argument M .
    • The denominator of the quotient will be a logarithm with base n and argument b .

Changing logarithmic expressions to expressions involving only natural logs

Change log 5 3 to a quotient of natural logarithms.

Because we will be expressing log 5 3 as a quotient of natural logarithms, the new base, n = e .

We rewrite the log as a quotient using the change-of-base formula. The numerator of the quotient will be the natural log with argument 3. The denominator of the quotient will be the natural log with argument 5.

log b M = ln M ln b log 5 3 = ln 3 ln 5
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Change log 0.5 8 to a quotient of natural logarithms.

ln 8 ln 0.5

Got questions? Get instant answers now!

Can we change common logarithms to natural logarithms?

Yes. Remember that log 9 means log 10 9 . So, log 9 = ln 9 ln 10 .

Using the change-of-base formula with a calculator

Evaluate log 2 ( 10 ) using the change-of-base formula with a calculator.

According to the change-of-base formula, we can rewrite the log base 2 as a logarithm of any other base. Since our calculators can evaluate the natural log, we might choose to use the natural logarithm, which is the log base e .

log 2 10 = ln 10 ln 2 Apply the change of base formula using base  e . 3.3219 Use a calculator to evaluate to 4 decimal places .
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Evaluate log 5 ( 100 ) using the change-of-base formula.

ln 100 ln 5 4.6051 1.6094 = 2.861

Got questions? Get instant answers now!

Access these online resources for additional instruction and practice with laws of logarithms.

Key equations

The Product Rule for Logarithms log b ( M N ) = log b ( M ) + log b ( N )
The Quotient Rule for Logarithms log b ( M N ) = log b M log b N
The Power Rule for Logarithms log b ( M n ) = n log b M
The Change-of-Base Formula log b M = log n M log n b           n > 0 , n 1 , b 1

Key concepts

  • We can use the product rule of logarithms to rewrite the log of a product as a sum of logarithms. See [link] .
  • We can use the quotient rule of logarithms to rewrite the log of a quotient as a difference of logarithms. See [link] .
  • We can use the power rule for logarithms to rewrite the log of a power as the product of the exponent and the log of its base. See [link] , [link] , and [link] .
  • We can use the product rule, the quotient rule, and the power rule together to combine or expand a logarithm with a complex input. See [link] , [link] , and [link] .
  • The rules of logarithms can also be used to condense sums, differences, and products with the same base as a single logarithm. See [link] , [link] , [link] , and [link] .
  • We can convert a logarithm with any base to a quotient of logarithms with any other base using the change-of-base formula. See [link] .
  • The change-of-base formula is often used to rewrite a logarithm with a base other than 10 and e as the quotient of natural or common logs. That way a calculator can be used to evaluate. See [link] .

Section exercises

Verbal

How does the power rule for logarithms help when solving logarithms with the form log b ( x n ) ?

Any root expression can be rewritten as an expression with a rational exponent so that the power rule can be applied, making the logarithm easier to calculate. Thus, log b ( x 1 n ) = 1 n log b ( x ) .

Got questions? Get instant answers now!

What does the change-of-base formula do? Why is it useful when using a calculator?

Got questions? Get instant answers now!

Algebraic

For the following exercises, expand each logarithm as much as possible. Rewrite each expression as a sum, difference, or product of logs.

log b ( 7 x 2 y )

log b ( 2 ) + log b ( 7 ) + log b ( x ) + log b ( y )

Got questions? Get instant answers now!

log b ( 13 17 )

log b ( 13 ) log b ( 17 )

Got questions? Get instant answers now!

ln ( 1 4 k )

k ln ( 4 )

Got questions? Get instant answers now!

For the following exercises, condense to a single logarithm if possible.

ln ( 7 ) + ln ( x ) + ln ( y )

ln ( 7 x y )

Got questions? Get instant answers now!

log 3 ( 2 ) + log 3 ( a ) + log 3 ( 11 ) + log 3 ( b )

Got questions? Get instant answers now!

log b ( 28 ) log b ( 7 )

log b ( 4 )

Got questions? Get instant answers now!

ln ( a ) ln ( d ) ln ( c )

Got questions? Get instant answers now!

log b ( 1 7 )

log b ( 7 )

Got questions? Get instant answers now!

For the following exercises, use the properties of logarithms to expand each logarithm as much as possible. Rewrite each expression as a sum, difference, or product of logs.

log ( x 15 y 13 z 19 )

15 log ( x ) + 13 log ( y ) 19 log ( z )

Got questions? Get instant answers now!

ln ( a −2 b −4 c 5 )

Got questions? Get instant answers now!

log ( x 3 y 4 )

3 2 log ( x ) 2 log ( y )

Got questions? Get instant answers now!

log ( x 2 y 3 x 2 y 5 3 )

8 3 log ( x ) + 14 3 log ( y )

Got questions? Get instant answers now!

For the following exercises, condense each expression to a single logarithm using the properties of logarithms.

log ( 2 x 4 ) + log ( 3 x 5 )

Got questions? Get instant answers now!

ln ( 6 x 9 ) ln ( 3 x 2 )

ln ( 2 x 7 )

Got questions? Get instant answers now!

2 log ( x ) + 3 log ( x + 1 )

Got questions? Get instant answers now!

log ( x ) 1 2 log ( y ) + 3 log ( z )

log ( x z 3 y )

Got questions? Get instant answers now!

4 log 7 ( c ) + log 7 ( a ) 3 + log 7 ( b ) 3

Got questions? Get instant answers now!

For the following exercises, rewrite each expression as an equivalent ratio of logs using the indicated base.

log 7 ( 15 ) to base e

log 7 ( 15 ) = ln ( 15 ) ln ( 7 )

Got questions? Get instant answers now!

log 14 ( 55.875 ) to base 10

Got questions? Get instant answers now!

For the following exercises, suppose log 5 ( 6 ) = a and log 5 ( 11 ) = b . Use the change-of-base formula along with properties of logarithms to rewrite each expression in terms of a and b . Show the steps for solving.

log 11 ( 5 )

log 11 ( 5 ) = log 5 ( 5 ) log 5 ( 11 ) = 1 b

Got questions? Get instant answers now!

log 11 ( 6 11 )

log 11 ( 6 11 ) = log 5 ( 6 11 ) log 5 ( 11 ) = log 5 ( 6 ) log 5 ( 11 ) log 5 ( 11 ) = a b b = a b 1

Got questions? Get instant answers now!

Numeric

For the following exercises, use properties of logarithms to evaluate without using a calculator.

log 3 ( 1 9 ) 3 log 3 ( 3 )

Got questions? Get instant answers now!

6 log 8 ( 2 ) + log 8 ( 64 ) 3 log 8 ( 4 )

3

Got questions? Get instant answers now!

2 log 9 ( 3 ) 4 log 9 ( 3 ) + log 9 ( 1 729 )

Got questions? Get instant answers now!

For the following exercises, use the change-of-base formula to evaluate each expression as a quotient of natural logs. Use a calculator to approximate each to five decimal places.

log 1 2 ( 4.7 )

2.23266

Got questions? Get instant answers now!

Extensions

Use the product rule for logarithms to find all x values such that log 12 ( 2 x + 6 ) + log 12 ( x + 2 ) = 2. Show the steps for solving.

Got questions? Get instant answers now!

Use the quotient rule for logarithms to find all x values such that log 6 ( x + 2 ) log 6 ( x 3 ) = 1. Show the steps for solving.

x = 4 ; By the quotient rule: log 6 ( x + 2 ) log 6 ( x 3 ) = log 6 ( x + 2 x 3 ) = 1.

Rewriting as an exponential equation and solving for x :

6 1 = x + 2 x 3 0 = x + 2 x 3 6 0 = x + 2 x 3 6 ( x 3 ) ( x 3 ) 0 = x + 2 6 x + 18 x 3 0 = x 4 x 3 x = 4

Checking, we find that log 6 ( 4 + 2 ) log 6 ( 4 3 ) = log 6 ( 6 ) log 6 ( 1 ) is defined, so x = 4.

Got questions? Get instant answers now!

Can the power property of logarithms be derived from the power property of exponents using the equation b x = m ? If not, explain why. If so, show the derivation.

Got questions? Get instant answers now!

Prove that log b ( n ) = 1 log n ( b ) for any positive integers b > 1 and n > 1.

Let b and n be positive integers greater than 1. Then, by the change-of-base formula, log b ( n ) = log n ( n ) log n ( b ) = 1 log n ( b ) .

Got questions? Get instant answers now!

Does log 81 ( 2401 ) = log 3 ( 7 ) ? Verify the claim algebraically.

Got questions? Get instant answers now!

Questions & Answers

what is mutation
Janga Reply
what is a cell
Sifune Reply
how is urine form
Sifune
what is antagonism?
mahase Reply
classification of plants, gymnosperm features.
Linsy Reply
what is the features of gymnosperm
Linsy
how many types of solid did we have
Samuel Reply
what is an ionic bond
Samuel
What is Atoms
Daprince Reply
what is fallopian tube
Merolyn
what is bladder
Merolyn
what's bulbourethral gland
Eduek Reply
urine is formed in the nephron of the renal medulla in the kidney. It starts from filtration, then selective reabsorption and finally secretion
onuoha Reply
State the evolution relation and relevance between endoplasmic reticulum and cytoskeleton as it relates to cell.
Jeremiah
what is heart
Konadu Reply
how is urine formed in human
Konadu
how is urine formed in human
Rahma
what is the diference between a cavity and a canal
Pelagie Reply
what is the causative agent of malaria
Diamond
malaria is caused by an insect called mosquito.
Naomi
Malaria is cause by female anopheles mosquito
Isaac
Malaria is caused by plasmodium Female anopheles mosquitoe is d carrier
Olalekan
a canal is more needed in a root but a cavity is a bad effect
Commander
what are pathogens
Don Reply
In biology, a pathogen (Greek: πάθος pathos "suffering", "passion" and -γενής -genēs "producer of") in the oldest and broadest sense, is anything that can produce disease. A pathogen may also be referred to as an infectious agent, or simply a germ. The term pathogen came into use in the 1880s.[1][2
Zainab
A virus
Commander
Definition of respiration
Muhsin Reply
respiration is the process in which we breath in oxygen and breath out carbon dioxide
Achor
how are lungs work
Commander
where does digestion begins
Achiri Reply
in the mouth
EZEKIEL
what are the functions of follicle stimulating harmones?
Rashima Reply
stimulates the follicle to release the mature ovum into the oviduct
Davonte
what are the functions of Endocrine and pituitary gland
Chinaza
endocrine secrete hormone and regulate body process
Achor
while pituitary gland is an example of endocrine system and it's found in the Brain
Achor
what's biology?
Egbodo Reply
Biology is the study of living organisms, divided into many specialized field that cover their morphology, physiology,anatomy, behaviour,origin and distribution.
Lisah
biology is the study of life.
Alfreda
Biology is the study of how living organisms live and survive in a specific environment
Sifune
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
Practice Key Terms 4

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, College algebra. OpenStax CNX. Feb 06, 2015 Download for free at https://legacy.cnx.org/content/col11759/1.3
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'College algebra' conversation and receive update notifications?

Ask