<< 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

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
Aislinn Reply
cm
tijani
what is titration
John Reply
what is physics
Siyaka Reply
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Jude Reply
Can you compute that for me. Ty
Jude
what is the dimension formula of energy?
David Reply
what is viscosity?
David
what is inorganic
emma Reply
what is chemistry
Youesf Reply
what is inorganic
emma
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
please, I'm a physics student and I need help in physics
Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
Krampah Reply
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
Sahid Reply
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
Ryan
what's motion
Maurice Reply
what are the types of wave
Maurice
answer
Magreth
progressive wave
Magreth
hello friend how are you
Muhammad Reply
fine, how about you?
Mohammed
hi
Mujahid
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
yasuo Reply
Who can show me the full solution in this problem?
Reofrir Reply
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, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

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

Ask