Does Repetitive Division Correspond to a Logarithm?
Date: 05/26/2008 at 19:20:08 From: Amir Subject: How to determine from division if a function is logarithmic Hello, I was in a class recently and it's been a few years since I studied exponents and logarithms but my interest in them never ceases! I understand that logs and exponents are inverse functions and how they correlate in that sense. I also understand that when you have a number and you continuously multiply it by itself this is considered exponential and I know what the graph of that looks like, e.g., f(x) = 2^x. Where I got a bit confused was when the teacher mentioned, "Well if we continuously divide this number then what do we have? A logarithmic function." So can we perceive logs in another way as this: since exponents are just the multiplication of numbers, and since logs are the inverse function, then the division of a number is similar to a logarithmic function. Basically, can we correlate that somehow? I haven't really been able to prove much of this idea that I have that correlates multiplication/division to exponents/logs. I've searched online but to no avail.
Date: 05/27/2008 at 09:12:54 From: Doctor Ian Subject: Re: How to determine from division if a function is logarithmic Hi Amir, This is a good question. Thanks for asking it. I've never really thought about it that way! Let's see what he might mean by it. Let's leave aside for the moment the continuous nature of both exponents and logarithms, e.g., we can raise any real base to any real exponent, e.g., pi^sqrt(2) = 5.0474969 (approximately) and just focus on integers. We can look at something like 3^4 and interpret that as multiply four copies of 3 = 3 * 3 * 3 * 3 = 81 Now, the first thing to note it that there are TWO ways to invert this process. If we know the exponent, we use a root to find the original base: 4___ \/ 81 = 3 And if we know the base, we can use the logarithm to find the exponent: log_3(81) = 4 We need two different inverses, because the operators aren't 'equal partners', as they are in multiplication. The root is answering the question: If we multiply four copies of some quantity and get a product of 81, what would that quantity be? The logarithm is answering the question: If we multiply some number of copies of 3 and get a product of 81, what would that number be? Now, suppose we decide we'd like to answer these numerically. That is, we'll make a guess, and try it, and use the error to come up with a better guess. In the case of the root, we might guess that the root is 4. Now we divide by 4, 3 times: 81 / 4 = 20.25 20.25 / 4 = 5.065 5.065 / 4 = 1.266 The result is too small, so we need to try a smaller guess. How about 3.1? 81 / 3.1 = 26.129 26.129 / 3.1 = 8.429 8.429 / 3.1 = 2.719 Still too small, but getting closer. (That is, 2.719 is closer to 3.1, than 1.266 was to 4. I need the initial and final numbers to be the same.) Let's try 3: 81 / 3 = 27 27 / 3 = 9 9 / 3 = 3 So 81 = 3 * 3 * 3 * 3 = 3^4 So a root is a succession of divisions, where we know how many times to divide (one less than the exponent), but not what to divide by (the base). Does that make sense? A logarithm would then deal with the other case: We know what to divide by (the base), but not how many times to divide. Again, we could do this by trial and error. Let's guess that 81 = 3^3, so we divide twice, with the target being to end at 3. 81 / 3 = 27 27 / 3 = 9 The result is too high, so we didn't divide enough times. How about 5? 81 / 3 = 27 27 / 3 = 9 9 / 3 = 3 3 / 3 = 1 Too low. We divided too many times. As we know, an exponent of 4-- right in between our two guesses--will be just right. :^D As I said, I don't want to get into the complexities involved when we move from integers to all real numbers. But do you see the relationship between exponents, roots, and logarithms, and how roots and logarithms can be understood as repeated divisions? Does this answer your question? - Doctor Ian, The Math Forum http://mathforum.org/dr.math/
Date: 07/14/2008 at 22:11:52 From: Amir Subject: Thank you (How to determine from division if a function is logarithmic) Thank you! You did a great job.
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