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I search for an answer but nobody really is answering or understanding what I mean by "significance" and I really don't know how to explain so I'm hoping somebody out there could give me a decent, abstract response.

But what exactly is the purpose of logarithms and what significance do they hold in higher mathematics?

Also, what is the significance of $(n-1)$? I just always see these in higher level equations but I want to better understand the purpose $(n-1)$ serve in mathematics.

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closed as unclear what you're asking by azarel, ShreevatsaR, Adriano, nbubis, Phira Aug 17 '13 at 12:24

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Logarithms are useful when, for example, multiplying as: log(x) + log(y) = log(xy). The significance of $(n-1)$ is that it is useful in situations where you want $n-1$... I guess... –  JohnWO Aug 14 '13 at 4:26
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$(n-1)$ isn't ringing a bell for anybody looking at this question. Could you give us an example of where this comes up? Also, logarithms are used fairly widely throughout mathematics. Could we narrow down "higher mathematics"? Are you talking about calculus? Number theory? Probability and statistics? If it's possible, it'd be better to start specific, then maybe generalize. –  Omnomnomnom Aug 14 '13 at 4:42
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Have you looked at the previous question on intuitive use of logarithms? –  Rahul Aug 14 '13 at 4:44
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Sometimes a cigar is just a cigar. –  Rahul Aug 14 '13 at 4:59
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The significane of $\ln n$ is of course that its reciprocal is a goodapproximation of th eprobability thet $n$ is prime :) –  Hagen von Eitzen Aug 14 '13 at 6:05

6 Answers 6

Logarithms are useful in that they can turn a multiplicative equation (Ex: $a = (b * c)$) into an additive equation ($\log a = \log b + \log c$). In the past, they have been used to speed up large multiplications by converting them into additions, which are simpler to do.
This feature of the logarithm is still useful, though less for speeding things up and more for changing the form of an equation to make it something that is easier to solve for whatever the unknown may be.

I'm not really sure what you mean by 'the significance of $(n - 1)$'. $n$ is often used in mathematics to represent an arbitrary natural number $(0, 1, 2, ...)$, and so $(n - 1)$ would represent one less than that number.

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lmao, thank you! Yeah, nobody seems to know what I mean. I know what (n - 1) is literally (a number minus 1) but I see it used in Probability, Physics equations & programming language as a step of a much larger job being done, but I can't seem to understand where it came from. I'd ask a professor of mine and they look at me like I'm speaking another language, haha. I'm sure it'll click but again, thank you for the response. –  dropout Aug 14 '13 at 4:52
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The main case that I can think of where you'd see (n - 1) used a lot in programming or in discrete maths is in induction. –  qaphla Aug 14 '13 at 5:03
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@dropout: Without an example, your question makes no sense. "n-1" comes when you want to subtract one from a number; it is bizarre to imagine that "where it comes from" would have a uniform answer across all situations. It is like asking where "n" comes from. –  ShreevatsaR Aug 14 '13 at 5:23
  1. As mentioned in the answers above, logarithms can turn a multiplication into addition by looking up values form log table. However, our electronic calculators are so powerful today that they make this way of computation obsolete. (But read on.)

  2. Logarithms are still useful in many areas. One example is:- Some functions (like $y = x^x$) cannot be differentiated directly, unless it has been converted into its corresponding logarithmic form first.

  3. One can consider mathematics is a sort of training or a game. A game has some rules. The players are required to (1) reach certain goals (2) obtain further (maybe more useful results) according to a list of rules. The list of formulas governing logarithms is one of such lists.

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Significance of logarithm is that it can change multiplications and divisions into additions
for example, $\log(a*b) = \log a + \log b$
If you have a log table at hand with you, you can easily find out the value of $\log a$ and $\log b$. then all you have to do is add them together, which is an easier operation than multiplication or division. You will get the result as the logarithm of the product that you were looking for, that is, you will get $\log(a*b)$. If you take the inverse log of this value (from an antilogarithm table) you will now get the value of $a*b$, without having to do any multiplication.

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Others explain logarithms very well. I won't repeat them.

As what is the significance of (n-1)?, I would like to answer it first by asking what is the significance of (n+1)?

In the sense of programming, $n + 1$ mean increment of $n$ by one, $n -1$ means decrement of $n$ by one. In plain English, if you are at position $n$, $n + 1$ means you go forward one step while $n - 1$ means you go backward by one step. It's simple as that. You don't want to drive a car which can only go forward.

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On n-1.

Terms such as n-1 and n+1 tend to appear when there is list of things we wish to refer to, or when we have a process which moves from one state to another, often in response to the flow of time, usually in discrete time periods.

In the situation where we have some property or process which evolves over time, such as the number of cells when something is growing in biology, the total amount of money to be paid under compound interest or the probability of picking a marble of a specific colour from a bag on the current attempt you will find terms such as n-1.

In such cases, to ] calculate the state of the system, we use the last state and some update which tells us how to move from the last state, n-1, to the current state, n.

You could of course view from a slightly different perspective and call the last known state, n, and the current state, n+1.

The distinction between these two naming conventions is reasonably arbitrary, and they essentially resolve to the same meaning.

Practically

You might want to start out by thinking of n-1 as representing the previous item in a sequence or list, or the last state of a system or process.

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If I understand you correctly, you are not asking for whether there is a significant use or benficiary application of the $\log$ but the intuition behind of its mathematical or even natural existence. This can be regarded as a natural philosophy question but could be answered in the mathematical context as well as the question of why natural numbers are well ordered.

The $\log$ function has a deep and profound natural intuition from mathematical point of view which results from the fact that it defines the reverse of an exponentiation.

$$a=b^c \rightarrow \log_b a=c$$

Even the multiplication/addition properties brought up in the answers below can be celebrated in this context of reverse of exponentiation. Hereof we also can lend the intuition for the well known $\log$-derivative $f'/f$ - to which connects to the intuition of exponential growth cross over all natural sciences for instance.

In this respect one can regard the significance of $\log$ to be that it directly points to the $valuation$ by reversing exponentiation. This enables a series of other applicative operative techniques such as addition versus multiplication.

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