<< Chapter < Page Chapter >> Page >

Finding the value of a common logarithm mentally

Evaluate y = log ( 1000 ) without using a calculator.

First we rewrite the logarithm in exponential form: 10 y = 1000. Next, we ask, “To what exponent must 10 be raised in order to get 1000?” We know

10 3 = 1000

Therefore, log ( 1000 ) = 3.

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

Evaluate y = log ( 1, 000, 000 ) .

log ( 1 , 000 , 000 ) = 6

Got questions? Get instant answers now!

Given a common logarithm with the form y = log ( x ) , evaluate it using a calculator.

  1. Press [LOG] .
  2. Enter the value given for x , followed by [ ) ] .
  3. Press [ENTER] .

Finding the value of a common logarithm using a calculator

Evaluate y = log ( 321 ) to four decimal places using a calculator.

  • Press [LOG] .
  • Enter 321 , followed by [ ) ] .
  • Press [ENTER] .

Rounding to four decimal places, log ( 321 ) 2.5065.

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

Evaluate y = log ( 123 ) to four decimal places using a calculator.

log ( 123 ) 2.0899

Got questions? Get instant answers now!

Rewriting and solving a real-world exponential model

The amount of energy released from one earthquake was 500 times greater than the amount of energy released from another. The equation 10 x = 500 represents this situation, where x is the difference in magnitudes on the Richter Scale. To the nearest thousandth, what was the difference in magnitudes?

We begin by rewriting the exponential equation in logarithmic form.

10 x = 500 log ( 500 ) = x Use the definition of the common log .

Next we evaluate the logarithm using a calculator:

  • Press [LOG] .
  • Enter 500 , followed by [ ) ] .
  • Press [ENTER] .
  • To the nearest thousandth, log ( 500 ) 2.699.

The difference in magnitudes was about 2.699.

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

The amount of energy released from one earthquake was 8,500 times greater than the amount of energy released from another. The equation 10 x = 8500 represents this situation, where x is the difference in magnitudes on the Richter Scale. To the nearest thousandth, what was the difference in magnitudes?

The difference in magnitudes was about 3.929.

Got questions? Get instant answers now!

Using natural logarithms

The most frequently used base for logarithms is e . Base e logarithms are important in calculus and some scientific applications; they are called natural logarithms . The base e logarithm, log e ( x ) , has its own notation, ln ( x ) .

Most values of ln ( x ) can be found only using a calculator. The major exception is that, because the logarithm of 1 is always 0 in any base, ln 1 = 0. For other natural logarithms, we can use the ln key that can be found on most scientific calculators. We can also find the natural logarithm of any power of e using the inverse property of logarithms.

Definition of the natural logarithm

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

For x > 0 ,

y = ln ( x )  is equivalent to  e y = x

We read ln ( x ) as, “the logarithm with base e of x ” or “the natural logarithm of x .

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

Since the functions y = e x and y = ln ( x ) are inverse functions, ln ( e x ) = x for all x and e = ln ( x ) x for x > 0.

Given a natural logarithm with the form y = ln ( x ) , evaluate it using a calculator.

  1. Press [LN] .
  2. Enter the value given for x , followed by [ ) ] .
  3. Press [ENTER] .

Questions & Answers

The sequence is {1,-1,1-1.....} has
amit Reply
circular region of radious
Kainat Reply
how can we solve this problem
Joel Reply
Sin(A+B) = sinBcosA+cosBsinA
Eseka Reply
Prove it
Eseka
Please prove it
Eseka
hi
Joel
June needs 45 gallons of punch. 2 different coolers. Bigger cooler is 5 times as large as smaller cooler. How many gallons in each cooler?
Arleathia Reply
find the sum of 28th term of the AP 3+10+17+---------
Prince Reply
I think you should say "28 terms" instead of "28th term"
Vedant
if sequence sn is a such that sn>0 for all n and lim sn=0than prove that lim (s1 s2............ sn) ke hole power n =n
SANDESH Reply
write down the polynomial function with root 1/3,2,-3 with solution
Gift Reply
if A and B are subspaces of V prove that (A+B)/B=A/(A-B)
Pream Reply
write down the value of each of the following in surd form a)cos(-65°) b)sin(-180°)c)tan(225°)d)tan(135°)
Oroke Reply
Prove that (sinA/1-cosA - 1-cosA/sinA) (cosA/1-sinA - 1-sinA/cosA) = 4
kiruba Reply
what is the answer to dividing negative index
Morosi Reply
In a triangle ABC prove that. (b+c)cosA+(c+a)cosB+(a+b)cisC=a+b+c.
Shivam Reply
give me the waec 2019 questions
Aaron Reply
the polar co-ordinate of the point (-1, -1)
Sumit Reply
Practice Key Terms 3

Get the best Algebra and trigonometry course in your pocket!





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