<< Chapter < Page Chapter >> Page >

We can express the relationship between logarithmic form and its corresponding exponential form as follows:

log b ( x ) = y b y = x , b > 0 , b 1

Note that the base b is always positive.

Because logarithm is a function, it is most correctly written as log b ( x ) , using parentheses to denote function evaluation, just as we would with f ( x ) . However, when the input is a single variable or number, it is common to see the parentheses dropped and the expression written without parentheses, as log b x . Note that many calculators require parentheses around the x .

We can illustrate the notation of logarithms as follows:

Notice that, comparing the logarithm function and the exponential function, the input and the output are switched. This means y = log b ( x ) and y = b x are inverse functions.

Definition of the logarithmic function

A logarithm    base b of a positive number x satisfies the following definition.

For x > 0 , b > 0 , b 1 ,

y = log b ( x )  is equivalent to  b y = x

where,

  • we read log b ( x ) as, “the logarithm with base b of x ” or the “log base b of x . "
  • the logarithm y is the exponent to which b must be raised to get x .

Also, since the logarithmic and exponential functions switch the x and y values, the domain and range of the exponential function are interchanged for the logarithmic function. Therefore,

  • the domain of the logarithm function with base b   is   ( 0 , ) .
  • the range of the logarithm function with base b   is   ( , ) .

Can we take the logarithm of a negative number?

No. Because the base of an exponential function is always positive, no power of that base can ever be negative. We can never take the logarithm of a negative number. Also, we cannot take the logarithm of zero. Calculators may output a log of a negative number when in complex mode, but the log of a negative number is not a real number.

Given an equation in logarithmic form log b ( x ) = y , convert it to exponential form.

  1. Examine the equation y = log b x and identify b , y , and x .
  2. Rewrite log b x = y as b y = x .

Converting from logarithmic form to exponential form

Write the following logarithmic equations in exponential form.

  1. log 6 ( 6 ) = 1 2
  2. log 3 ( 9 ) = 2

First, identify the values of b , y , and x . Then, write the equation in the form b y = x .

  1. log 6 ( 6 ) = 1 2

    Here, b = 6 , y = 1 2 , and   x = 6. Therefore, the equation log 6 ( 6 ) = 1 2 is equivalent to 6 1 2 = 6 .

  2. log 3 ( 9 ) = 2

    Here, b = 3 , y = 2 , and   x = 9. Therefore, the equation log 3 ( 9 ) = 2 is equivalent to 3 2 = 9.

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Write the following logarithmic equations in exponential form.

  1. log 10 ( 1, 000, 000 ) = 6
  2. log 5 ( 25 ) = 2
  1. log 10 ( 1 , 000 , 000 ) = 6 is equivalent to 10 6 = 1 , 000 , 000
  2. log 5 ( 25 ) = 2 is equivalent to 5 2 = 25
Got questions? Get instant answers now!

Converting from exponential to logarithmic form

To convert from exponents to logarithms, we follow the same steps in reverse. We identify the base b , exponent x , and output y . Then we write x = log b ( y ) .

Converting from exponential form to logarithmic form

Write the following exponential equations in logarithmic form.

  1. 2 3 = 8
  2. 5 2 = 25
  3. 10 4 = 1 10,000

First, identify the values of b , y , and x . Then, write the equation in the form x = log b ( y ) .

  1. 2 3 = 8

    Here, b = 2 , x = 3 , and y = 8. Therefore, the equation 2 3 = 8 is equivalent to log 2 ( 8 ) = 3.

  2. 5 2 = 25

    Here, b = 5 , x = 2 , and y = 25. Therefore, the equation 5 2 = 25 is equivalent to log 5 ( 25 ) = 2.

  3. 10 4 = 1 10,000

    Here, b = 10 , x = 4 , and y = 1 10,000 . Therefore, the equation 10 4 = 1 10,000 is equivalent to log 10 ( 1 10,000 ) = 4.

Got questions? Get instant answers now!
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 3

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?

Ask