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Logarithms(i hope this is in the right section) 
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#1
Nov1312, 09:43 PM

P: 190

Hey all,
so i finished logarithms a few weeks ago for mathematics/precalculus 12( also known as algebra II in the states i think ) and i got 100% on the test but don't feel i understand logs very much... i understand how to solve problems and the laws completely but i started reading a book called "In Pursuit Of The Unknown, 17 equations that changed the World" which has a chapter on the history of logarithms and i have came to the conclusion that i do not understand logarithms very much... ex: log4 = .6020599913 i understand its log base 10 therefore i know 10^.6020599913 = 4 1)but i don't understand why the word "log" is in there, what is the value of log? or is there a value? 2)what exactly does log4 mean? does it just mean 10 to the power of something = 4? 3)Also, will university math courses go more in depth with say logs, translations, arithmetic sequences and series and such? i have over 95% in this course yet i feel like i am not learning enough. There is no proofs, i have no idea why we use the formulas we do. Its simply just, this is how you do it or here's the formula, now solve the problem. it seems so juvenile. any help would be greatly appreciated 


#2
Nov1312, 09:55 PM

P: 2,949

did they do derivations to get the formulas? thats a form of proof.
Sorry, if you know this already but to reiterate: With respect to logarithms the log 4 is the same as log(4) ie log is a function like sqrt or sin/cos/tan. y=x^2 and you take the sqrt of both sides you get sqrt(y) = x or sqrt(y) = x so too with logs 4 = 10^x and taking the log of both sides you get log(4) = x logs reduce multiplications to additions and powers to multiplications. At one time before the advent of the personal handheld calculator, log measure was used on sliderules so that by adding two lengths you could multiply two numbers. 


#3
Nov1312, 10:07 PM

P: 190

wow that actually answered everything that was confusing me! thank you very much!
i just read about the use of sliderules in that book i mentioned haha usually he doesn't show us how to derive the equations, i usually try to do it by myself so i have a better understanding. If he does proofs its usually unclear on what exactly he is doing. Will Calculus I or Linear Algebra go more in depth? i really want to be prepared for university courses Thanks again! 


#4
Nov1312, 10:07 PM

P: 29

Logarithms(i hope this is in the right section)
2)Yes and you solve for that something. 3) I'm not sure about that. I agree with you. The current high school curriculum doesn't seem to cover a lot of proofs. But just google whatever proof you want to know, there's bound to be results. 


#5
Nov1312, 10:11 PM

P: 190

yeah i really get frustrated when i am just handed equations and told to use them... 


#6
Nov1312, 11:00 PM

P: 2,949

Most of the math classes you will take as an undergraduate will focus on the practical aspects of the math ie how to use them to solve problems, get answers and check that the answer is correct. When you take Abstract Algebra, Group Theory, and Topology then you begin to see the math behind the math.
For me the math helped me understand the physics I was doing although I was always left with the question of where does the math drop off and the physics begin. Most physics problems seemed to boundary value related, given some constraints and the equations of motion compute the actual motion. It wasn't until much later that I understood the math is the physics. The equations and the operations used are part of the theory. 


#7
Nov1412, 12:07 PM

P: 82

I remember I read somewhere, not sure whether it was a mathematics course or not, but that 'back in the day', navigators on board ships would be recording, well not sure what exactly, but the numbers they were recording were huge. So large in fact, that it started to become a problem just doing the recording. So they starting employing logarithms. I think of logarithms as giving exponents as solutions, but you probably already knew that. But since we typically need a far smaller exponent to base 10, than the number we are taking a logarithm of, these navigators used logarithms to get much smaller numbers useful for recording purposes. My memory of the details in this history are sketchy and this really doesn't answer your question I think, but a bit of history is always a great way to add life to a subject IMHO.
With your drive, I think you will simply love Calculus. I couldn't wait to get out of high school, and was stuck taking PreCalc in university for the first year, but I still remember my Calculus class as the first time I really saw beauty in math. 


#8
Nov1412, 01:59 PM

Mentor
P: 21,262

The primary reason for the development of logarithms was to make it easier to do multiplication and division. The basic ideas here are these properties of logs: log(ab) = log(a) + log(b) log(a/b) = log(a)  log(b) For example, if you need to calculate 63/30 and calculators haven't yet been invented, you can figure this out with long division. On the other hand, if you know that log(63) ≈ 1.7993 and log(30) ≈ 1.4771, then log(63)  log(30) ≈ .3222, which is the log(63/30). Finding the antilog of .3222 (i.e., finding 10^{.3222} gives you the answer, which is 2.1. Since I'm assuming that this calculation is taking place before the advent of calculators, you wouldn't be able to directly calculate 10^{.3222}, but the same table of logs that gave you log(63) and log(30) could be used to give you the antilog of .3222. 


#9
Nov1412, 03:06 PM

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#10
Nov1412, 03:16 PM

P: 82

Sorry, I wasn't trying to imply that early ship navigators invented logarithms, just giving one example of where they were used early on and how history can add depth to a subject's understanding.



#11
Nov1412, 03:46 PM

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#12
Nov1412, 04:35 PM

P: 2,949

Ship navigation did use logs for determine vessel speed:
http://en.wikipedia.org/wiki/Chip_log but maybe not the logs you were you're thinking of :) 


#13
Nov1412, 05:41 PM

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What he developed was an improvement, but not quite what he was hoping for. His system used a base of 1/e and was developed using a geometric method (a logical enough train of thought, since trigonometry is related to geometry). Henry Briggs figured out how to get from Napier's method to the real goal  a system using base 10 logs (also known as common logs). While, technically, Briggs's base 10 logs achieved the goal sought by Napier, it turns out e is a pretty important constant in itself (perhaps more important than just finding an easier way to do multiplication and division) and Napier's method was later revised to use base e instead of 1/e. Both systems have a very close relationship. In fact, natural logs (ln) are 2.30 times greater than base 10 logs (log). (Actually, 2.30258509+ times greater, but who needs more than 3 significant digits.) Napier also developed the first mechanical calculators: Napier's bones. Today, we make little kids memorize Napier's bones, except the teacher usually calls them multiplication tables so as not to freak out the little kids. 


#14
Nov1412, 06:25 PM

P: 82




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