Log, Ln, e --The World of Logarithms/Problems and Applications

Graph of z = Re(ln(x+iy)), selfmade with MuPad/ Sometimes we wonder if... the world can be equated to God's secret formulae...?
Graph of z = Re(ln(x+iy)), selfmade with MuPad/ Sometimes we wonder if... the world can be equated to God's secret formulae...? | Source

If you've ever seen this rover "Curiosity" that landed on Mars on August 6, 2012, chances are, NASA made use of complex binary algorithms. Let's put it simply this way: the rover was already going down on Martian surface with critical tasks of survival; last second decision making was handed by powerful computers, that put into practice that yes/no input/output.

Of course, we as general public/viewers saw it as another historical event, but behind all that awesomeness there was math, and logarithms for sure.


Just considered HubPages for a second. Those buttons for flagging, voting up/down are essentially handled by logarithmic formulae. There is an intrinsic association between logarithms and algorithms. The whole math involved in it can take us back to those flowcharts that involved commands like, If, GoTo, loops and of course "start" and "end." We are writing a more popular view of this torture of math. In real life we talk about getting input and delivering an output. Let's see what we have store:

Portrait of John Napier (1550-1617), the inventor of logarithms
Portrait of John Napier (1550-1617), the inventor of logarithms | Source

What The Heck is This Number "e"?

If you ask Mark Zuckerberg, you will be asking about a God for him. Is a number he needed in his algorithmic ideas to start FaceBook. Silicon Valley think of this number as magical, but what is it?

e= 2.7182

Was named Euler's constant in honor of Leonhard Euler (1707-1787), a Swiss Mathematician that loved his math like us.

If you want a more accurate number here it is

e=2.71828182845904523536028747135266249775724709369995.

But, what is it, Lord?

Is a constant number obtained from this sumatory : Σ (1 + 1/n)n for n=∞ or infinite.

It was chosen as the base for napiers natural logarithm ln with base e.

An example of this sum of infinite series is shown below.

Source
Source

Logarithm : Definition

John Napier introduced the name and its definition on His work, Mirifici Logarithmorum Canonis Descriptio (1614). Back then you had to write everything in Latin, and carefully spell the name of the King of the day whom the written piece would be dedicated to, with all kind of silly eulogies, sort of kissing their butt... but let's go straight to business:

Logarithm - ROUGH DEFINITION

The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number. For instance, you have the number 10,000 and you have a base of 10. You want to find the number x where:

10x =10,000

Common sense will indicate that x=4, but mathematicians wanted to make it look better and more academic... oh Gosh! Please, Sir John Napier and Monsieur Euler!

logarithmorum10 (10,000) = 4 (this nomenclature was understood in 1614-1615)

Right away it was shortened to "log"

Today, we recognize it better like this:

log10 (10,000) = 4

Napier, found that e=2.7182 was a more natural base than our base of 10 (100, 1000, 10,000)


LN is born

Smart enough, Napier squeezed his last name in the natural logarithm 'ln' (logarithmun naturalis). Why is better than 10? Well, life is based on natural increments. If you borrow money from the bank, the interest yielded will be based on this powerful number--Sort of π for "Geo" and "Trigo." I see those young kids saying..."ew! Now Lord is getting on our nerves!"

Life is not based in rigid numbers of 10, 20 or 30's. Don't want to go deeper, because then our brain will blow, before the end of this article. So help me God!



Source

Practical examples:

Solving algebraic equations involving logarithms

We are skipping some hard math and go straight to the point in here: Math can be harder enough if you follow your teacher's somber archaic methods.

We want to make sure we know this logarithm identities for further use.

log(a*b) = log(a) + log(b)

log(a/b) = log(a) - log(b)

log(ab) = b*log(a)

log(1) = 0 because 100=1 here and in China. Please, math is not a joke!


Any base to the power of the logarithm with the same base will cancel the base and the logarithm. This is the beauty:

  • 10 log10(x) = x

  • e ln(x) = x

EXAMPLE 1

log(5x+1) - log(x-5) = log(20) (base of 10)

If we make use of the trick of the relation : 10 log10(x) = x then we are getting somewhere:

Let's shortened the equation into an easier one

log((5x+1)/(x-5)) =log(20)

10 log(5x+1/X-5) = 10 log(20)



(5x-1)/(x-5) = 20 ( please Josh! You didn't pay attention? And you Jon?)

Now we have a more easier equation that happens to be linear (I know Miss Olive...I know! bear with us.)

5x-1=20(x-5)

5x-1=20x -100

moving x to the right side:

20x-5x=100-1

15x=99 x=99/15 x=6.6



Example 2

e(9x-5) - e (2x-1) =45

By using the relation: ln e x = x, we get rid of the monster equation and solve it!

lne(9x-5) - lne (2x-1) = ln (45)

ln(9x-5)/(2x-1)=ln(45) by simplifying we get:

(9x-5)/(2x-1)=ln(45) ; lets call ln(45)=k, by going to the natural log tables:

ln(45)= 3.8066624898 or 3.8, k=3.8

Solving the equation:

9x-5 =k(2x-1)

9x-5= 2xk-k

9x-2xk=5-k (almost done Josh!)

x(9-2k)=5-k

x=(5-k)/(9-2k) replacing k=3.8= ln(45)

we find x:

x=5-3.8/[9-2(3.8)]

x=1.2/(9-7.6)

x=1.2/1.4

x=0.857





We study this logarithms for nothing?

Certainly, if you want to be a scientist or chemist you will see it at the pH of any solution. Or perhaps at the Richter scale quake measurement magnitude formula; or behind the HP scoring...? You don't have to kill yourself by going any further. We are who we are. We all have unique qualities, and we all fit in the equation of life. But at least you got the hang of it, right?

This is not us....Just in case Tammy. Kelly, Ardie or Lathing ask...!

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Comments 11 comments

josh3418 profile image

josh3418 4 years ago from Pennsylvania

Joseph,

Excellent hub on logarithms! This one is a little over my head, because this was never covered in complete detail when I went to school. Thanks for sharing this awesome information clearly Joseph! You rock! Have a great rest of your day!


Janine Huldie profile image

Janine Huldie 4 years ago from New York, New York

Joseph, I actually had to teach this topic to one of my classes my first year of teaching and can tell you kids get so last with all the rules, example how multiplication with logs are really addition and so forth. I like the way ou approached this though and have pinned for further reference. Nice job and have voted and shared too!


tammyswallow profile image

tammyswallow 4 years ago from North Carolina

I was just discussing this subject of solving for e and logarithms with a friend over coffee last week and low and behold, the Mars Rover found something. Yeah...... Great hub even if it is greek to me.


tsmog profile image

tsmog 4 years ago from Escondido, CA

Well, I agree, applauding this expose of the how more than the what. Very well done lord (J0seph too) and thank you for a trip in that time machine. Even though I really couldn't do it today, I remember when a fallen log and all those signs led one down curvy roads of discovery.

I remember when I headed into analytical geometry proud of the slide rule given me by grand dad. That slide rule led him through Stanford around the 20's, me through high school in the 70's, then stolen it seems. Yet, today it would be used by a preschooler, nahhhhhhhhhh, they have smart phones and I-pads.

The sad part is I have reminiscent thoughts how to use that slider rule, always an answer it provided, yet a smart phone, not owning one, frankly scares the blank out of me - lol. The oddity is you have just shared the secret of them there gadgets. With that tiny bit of magnificent knowledge shared here, why like you said I could go to Mars or farther with a tad bit of time.

Thank you for sharing these words and 'your' time putting them together causing simplicity from complexity.

tim


Faith Reaper profile image

Faith Reaper 4 years ago from southern USA

lord, you had me (I mean you lost me) at Log, Ln, e - Ha. I did well in Algebra, but then when Advanced Algebra came -oh well, I guess it was that type of teacher you were speaking about - Ha At least, that is what I am saying. My little brain blew a fuse trying to understand this one, Way Over My Head - I even graduated with honors and belonged to the National Honor Society - must not mean much. Ha. God bless brains like yours!!! In His Love, Faith Reaper


Lord De Cross profile image

Lord De Cross 4 years ago Author

Josh, Janine and Tammy. We write this math hubs in order to reach our struggling students. Is sad how teachers pass on the burden of study to their students. We all went through that and sometimes we don't want to see those Sin or Cotang signs. Thanks for leaving meaninful comments.


Lord De Cross profile image

Lord De Cross 4 years ago Author

Hi Tim, catching up with comments in here. Simplicity have made wonders and it always be. Our brain will bw attracted to simple solutions. Only the chosen ones, will come up with simple answer to complex problems. I know what you mean about the sliding rule where given log2 and you try to lign it up with, for instance log3, you will get log6=log2x3= log2 +log3. Iphones are nothing nowadays. Just thinh of tridimensional FB gadgetry in the middle of your living room. The holographic technology is here, but still to expensive for the masses.


A.A. Zavala profile image

A.A. Zavala 4 years ago from Texas

Absolutely fascinating. Thank you for the refresher, now going to get some excederin for the aftermath. I have dumbed down in the last few years. I blame the media, and the media blamers. Thank you for sharing.


Lord De Cross profile image

Lord De Cross 4 years ago Author

Catching up with your comment Agustine. Is not that we are dumb, things are easier than before. We used to make those long divisions on a piece of paper, now we have computers, cellphones and even a calculator. Thanks Buddy!


B. Leekley profile image

B. Leekley 4 years ago from Kalamazoo, Michigan, USA

I wish I could read math.


Lord De Cross profile image

Lord De Cross 4 years ago Author

B leekley, Thanks for reading us. You don't need to read math, you can use it everyday. Just use that brainpower with no end. Math is being taught easier online, and that is an advantage that we never had as kids.

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