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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

what is functions?
Angel Reply
A mathematical relation such that every input has only one out.
Spiro
yes..it is a relationo of orders pairs of sets one or more input that leads to a exactly one output.
Mubita
If the plane intersects the cone (either above or below) horizontally, what figure will be created?
Feemark Reply
can you not take the square root of a negative number
Sharon Reply
No because a negative times a negative is a positive. No matter what you do you can never multiply the same number by itself and end with a negative
lurverkitten
Actually you can. you get what's called an Imaginary number denoted by i which is represented on the complex plane. The reply above would be correct if we were still confined to the "real" number line.
Liam
Suppose P= {-3,1,3} Q={-3,-2-1} and R= {-2,2,3}.what is the intersection
Elaine Reply
can I get some pretty basic questions
Ama Reply
In what way does set notation relate to function notation
Ama
is precalculus needed to take caculus
Amara Reply
It depends on what you already know. Just test yourself with some precalculus questions. If you find them easy, you're good to go.
Spiro
the solution doesn't seem right for this problem
Mars Reply
what is the domain of f(x)=x-4/x^2-2x-15 then
Conney Reply
x is different from -5&3
Seid
All real x except 5 and - 3
Spiro
***youtu.be/ESxOXfh2Poc
Loree
how to prroved cos⁴x-sin⁴x= cos²x-sin²x are equal
jeric Reply
Don't think that you can.
Elliott
By using some imaginary no.
Tanmay
how do you provided cos⁴x-sin⁴x = cos²x-sin²x are equal
jeric Reply
What are the question marks for?
Elliott
Someone should please solve it for me Add 2over ×+3 +y-4 over 5 simplify (×+a)with square root of two -×root 2 all over a multiply 1over ×-y{(×-y)(×+y)} over ×y
Abena Reply
For the first question, I got (3y-2)/15 Second one, I got Root 2 Third one, I got 1/(y to the fourth power) I dont if it's right cause I can barely understand the question.
Is under distribute property, inverse function, algebra and addition and multiplication function; so is a combined question
Abena
find the equation of the line if m=3, and b=-2
Ashley Reply
graph the following linear equation using intercepts method. 2x+y=4
Ashley
how
Wargod
what?
John
ok, one moment
UriEl
how do I post your graph for you?
UriEl
it won't let me send an image?
UriEl
also for the first one... y=mx+b so.... y=3x-2
UriEl
y=mx+b you were already given the 'm' and 'b'. so.. y=3x-2
Tommy
Please were did you get y=mx+b from
Abena
y=mx+b is the formula of a straight line. where m = the slope & b = where the line crosses the y-axis. In this case, being that the "m" and "b", are given, all you have to do is plug them into the formula to complete the equation.
Tommy
thanks Tommy
Nimo
0=3x-2 2=3x x=3/2 then . y=3/2X-2 I think
Given
co ordinates for x x=0,(-2,0) x=1,(1,1) x=2,(2,4)
neil
"7"has an open circle and "10"has a filled in circle who can I have a set builder notation
Fiston Reply
Where do the rays point?
Spiro
x=-b+_Гb2-(4ac) ______________ 2a
Ahlicia Reply
I've run into this: x = r*cos(angle1 + angle2) Which expands to: x = r(cos(angle1)*cos(angle2) - sin(angle1)*sin(angle2)) The r value confuses me here, because distributing it makes: (r*cos(angle2))(cos(angle1) - (r*sin(angle2))(sin(angle1)) How does this make sense? Why does the r distribute once
Carlos Reply
so good
abdikarin
this is an identity when 2 adding two angles within a cosine. it's called the cosine sum formula. there is also a different formula when cosine has an angle minus another angle it's called the sum and difference formulas and they are under any list of trig identities
Brad
strategies to form the general term
carlmark
consider r(a+b) = ra + rb. The a and b are the trig identity.
Mike
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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