<< Chapter < Page Chapter >> Page >

Write the following exponential equations in logarithmic form.

  1. 3 2 = 9
  2. 5 3 = 125
  3. 2 1 = 1 2
  1. 3 2 = 9 is equivalent to log 3 ( 9 ) = 2
  2. 5 3 = 125 is equivalent to log 5 ( 125 ) = 3
  3. 2 1 = 1 2 is equivalent to log 2 ( 1 2 ) = 1
Got questions? Get instant answers now!

Evaluating logarithms

Knowing the squares, cubes, and roots of numbers allows us to evaluate many logarithms mentally. For example, consider log 2 8. We ask, “To what exponent must 2 be raised in order to get 8?” Because we already know 2 3 = 8 , it follows that log 2 8 = 3.

Now consider solving log 7 49 and log 3 27 mentally.

  • We ask, “To what exponent must 7 be raised in order to get 49?” We know 7 2 = 49. Therefore, log 7 49 = 2
  • We ask, “To what exponent must 3 be raised in order to get 27?” We know 3 3 = 27. Therefore, log 3 27 = 3

Even some seemingly more complicated logarithms can be evaluated without a calculator. For example, let’s evaluate log 2 3 4 9 mentally.

  • We ask, “To what exponent must 2 3 be raised in order to get 4 9 ? ” We know 2 2 = 4 and 3 2 = 9 , so ( 2 3 ) 2 = 4 9 . Therefore, log 2 3 ( 4 9 ) = 2.

Given a logarithm of the form y = log b ( x ) , evaluate it mentally.

  1. Rewrite the argument x as a power of b : b y = x .
  2. Use previous knowledge of powers of b identify y by asking, “To what exponent should b be raised in order to get x ?

Solving logarithms mentally

Solve y = log 4 ( 64 ) without using a calculator.

First we rewrite the logarithm in exponential form: 4 y = 64. Next, we ask, “To what exponent must 4 be raised in order to get 64?”

We know

4 3 = 64

Therefore,

log ( 64 ) 4 = 3
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Solve y = log 121 ( 11 ) without using a calculator.

log 121 ( 11 ) = 1 2 (recalling that 121 = ( 121 ) 1 2 = 11 )

Got questions? Get instant answers now!

Evaluating the logarithm of a reciprocal

Evaluate y = log 3 ( 1 27 ) without using a calculator.

First we rewrite the logarithm in exponential form: 3 y = 1 27 . Next, we ask, “To what exponent must 3 be raised in order to get 1 27 ?

We know 3 3 = 27 , but what must we do to get the reciprocal, 1 27 ? Recall from working with exponents that b a = 1 b a . We use this information to write

3 3 = 1 3 3 = 1 27

Therefore, log 3 ( 1 27 ) = 3.

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

Evaluate y = log 2 ( 1 32 ) without using a calculator.

log 2 ( 1 32 ) = 5

Got questions? Get instant answers now!

Using common logarithms

Sometimes we may see a logarithm written without a base. In this case, we assume that the base is 10. In other words, the expression log ( x ) means log 10 ( x ) . We call a base-10 logarithm a common logarithm . Common logarithms are used to measure the Richter Scale mentioned at the beginning of the section. Scales for measuring the brightness of stars and the pH of acids and bases also use common logarithms.

Definition of the common logarithm

A common logarithm    is a logarithm with base 10. We write log 10 ( x ) simply as log ( x ) . The common logarithm of a positive number x satisfies the following definition.

For x > 0 ,

y = log ( x )  is equivalent to  10 y = x

We read log ( x ) as, “the logarithm with base 10 of x ” or “log base 10 of x .

The logarithm y is the exponent to which 10 must be raised to get x .

Given a common logarithm of the form y = log ( x ) , evaluate it mentally.

  1. Rewrite the argument x as a power of 10 : 10 y = x .
  2. Use previous knowledge of powers of 10 to identify y by asking, “To what exponent must 10 be raised in order to get x ?

Questions & Answers

how does Neisseria cause meningitis
Nyibol Reply
what is microbiologist
Muhammad Reply
what is errata
Muhammad
is the branch of biology that deals with the study of microorganisms.
Ntefuni Reply
What is microbiology
Mercy Reply
studies of microbes
Louisiaste
when we takee the specimen which lumbar,spin,
Ziyad Reply
How bacteria create energy to survive?
Muhamad Reply
Bacteria doesn't produce energy they are dependent upon their substrate in case of lack of nutrients they are able to make spores which helps them to sustain in harsh environments
_Adnan
But not all bacteria make spores, l mean Eukaryotic cells have Mitochondria which acts as powerhouse for them, since bacteria don't have it, what is the substitution for it?
Muhamad
they make spores
Louisiaste
what is sporadic nd endemic, epidemic
Aminu Reply
the significance of food webs for disease transmission
Abreham
food webs brings about an infection as an individual depends on number of diseased foods or carriers dully.
Mark
explain assimilatory nitrate reduction
Esinniobiwa Reply
Assimilatory nitrate reduction is a process that occurs in some microorganisms, such as bacteria and archaea, in which nitrate (NO3-) is reduced to nitrite (NO2-), and then further reduced to ammonia (NH3).
Elkana
This process is called assimilatory nitrate reduction because the nitrogen that is produced is incorporated in the cells of microorganisms where it can be used in the synthesis of amino acids and other nitrogen products
Elkana
Examples of thermophilic organisms
Shu Reply
Give Examples of thermophilic organisms
Shu
advantages of normal Flora to the host
Micheal Reply
Prevent foreign microbes to the host
Abubakar
they provide healthier benefits to their hosts
ayesha
They are friends to host only when Host immune system is strong and become enemies when the host immune system is weakened . very bad relationship!
Mark
what is cell
faisal Reply
cell is the smallest unit of life
Fauziya
cell is the smallest unit of life
Akanni
ok
Innocent
cell is the structural and functional unit of life
Hasan
is the fundamental units of Life
Musa
what are emergency diseases
Micheal Reply
There are nothing like emergency disease but there are some common medical emergency which can occur simultaneously like Bleeding,heart attack,Breathing difficulties,severe pain heart stock.Hope you will get my point .Have a nice day ❣️
_Adnan
define infection ,prevention and control
Innocent
I think infection prevention and control is the avoidance of all things we do that gives out break of infections and promotion of health practices that promote life
Lubega
Heyy Lubega hussein where are u from?
_Adnan
en français
Adama
which site have a normal flora
ESTHER Reply
Many sites of the body have it Skin Nasal cavity Oral cavity Gastro intestinal tract
Safaa
skin
Asiina
skin,Oral,Nasal,GIt
Sadik
How can Commensal can Bacteria change into pathogen?
Sadik
How can Commensal Bacteria change into pathogen?
Sadik
all
Tesfaye
by fussion
Asiina
what are the advantages of normal Flora to the host
Micheal
what are the ways of control and prevention of nosocomial infection in the hospital
Micheal
what is inflammation
Shelly Reply
part of a tissue or an organ being wounded or bruised.
Wilfred
what term is used to name and classify microorganisms?
Micheal Reply
Binomial nomenclature
adeolu
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, 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