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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.

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

log ( 1 , 000 , 000 ) = 6

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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.

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

log ( 123 ) 2.0899

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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.

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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.

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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

how to understand calculus?
Jenica Reply
Hey I am new to precalculus, and wanted clarification please on what sine is as I am floored by the terms in this app? I don't mean to sound stupid but I have only completed up to college algebra.
rachel Reply
I don't know if you are looking for a deeper answer or not, but the sine of an angle in a right triangle is the length of the opposite side to the angle in question divided by the length of the hypotenuse of said triangle.
Marco
can you give me sir tips to quickly understand precalculus. Im new too in that topic. Thanks
Jenica
if you remember sine, cosine, and tangent from geometry, all the relationships are the same but they use x y and r instead (x is adjacent, y is opposite, and r is hypotenuse).
Natalie
the standard equation of the ellipse that has vertices (0,-4)&(0,4) and foci (0, -15)&(0,15) it's standard equation is x^2 + y^2/16 =1 tell my why is it only x^2? why is there no a^2?
Reena Reply
what is foci?
Reena Reply
This term is plural for a focus, it is used for conic sections. For more detail or other math questions. I recommend researching on "Khan academy" or watching "The Organic Chemistry Tutor" YouTube channel.
Chris
how to determine the vertex,focus,directrix and axis of symmetry of the parabola by equations
Bryssen Reply
i want to sure my answer of the exercise
meena Reply
what is the diameter of(x-2)²+(y-3)²=25
Den Reply
how to solve the Identity ?
Barcenas Reply
what type of identity
Jeffrey
Confunction Identity
Barcenas
how to solve the sums
meena
hello guys
meena
For each year t, the population of a forest of trees is represented by the function A(t) = 117(1.029)t. In a neighboring forest, the population of the same type of tree is represented by the function B(t) = 86(1.025)t.
Shakeena Reply
by how many trees did forest "A" have a greater number?
Shakeena
32.243
Kenard
how solve standard form of polar
Rhudy Reply
what is a complex number used for?
Drew Reply
It's just like any other number. The important thing to know is that they exist and can be used in computations like any number.
Steve
I would like to add that they are used in AC signal analysis for one thing
Scott
Good call Scott. Also radar signals I believe.
Steve
They are used in any profession where the phase of a waveform has to be accounted for in the calculations. Imagine two electrical signals in a wire that are out of phase by 90°. At some times they will interfere constructively, others destructively. Complex numbers simplify those equations
Tim
Is there any rule we can use to get the nth term ?
Anwar Reply
how do you get the (1.4427)^t in the carp problem?
Gabrielle Reply
A hedge is contrusted to be in the shape of hyperbola near a fountain at the center of yard.the hedge will follow the asymptotes y=x and y=-x and closest distance near the distance to the centre fountain at 5 yards find the eqution of the hyperbola
ayesha Reply
A doctor prescribes 125 milligrams of a therapeutic drug that decays by about 30% each hour. To the nearest hour, what is the half-life of the drug?
Sandra Reply
Practice Key Terms 3

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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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