## Can someone explain logarithms?

Seeing all of the other math Instructables on here, I was surprised not see one on logarithms. I just want to learn beyond my mundane curriculum. Thanks.

Seeing all of the other math Instructables on here, I was surprised not see one on logarithms. I just want to learn beyond my mundane curriculum. Thanks.

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Posted:

Mar 1, 2009

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_{b}(N) is that number which, when used to exponentiate b, gives you N. However, I think you're asking, "what are logarithms good for?" Nowadays, not much unless you are a physicist specializing in either thermodynamics or QCD.In the old days, before calculators, logarithms were a way to allow you to do fairly complex arithmetic with nothing but addition and subtraction. For multiplication, there is a rule for exponents: b

^{x}×b^{y}= b^{x+y}. So if you have two numbers N and M which you want to multiply, then you can look up the logarithms, and use log(N)+log(M) = log(N×M). Similarly, for division you have b^{x}/b^{y}= b^{x-y}, and you can use log(N)-log(M) = log(N/M). If you have some complex calculation, you can do everything in terms of logs, and only look up the exponential ("inverse log") at the very end.then you can look up the logarithms_{hee hee, log tables....}In the old days, before calculators, logarithms were a way to allow you to do fairly complex arithmetic with nothing but addition and subtraction.In a very real sense, you can't really do anything to a number except add to it or subtract from it. The rest of math is just fast and fancy ways of doing this.

I'm not fully convinced that your statement is universally correct. Consider the operation of raising to a non-integer power (which is the whole point of logs, after all). Integer powers are equivalent to multiplication (n

^{3 = n×n×n), and multiplication is equivalent to repeated addition. }But if you raise a number to a floating-point power, how do you break that down into addition and subtraction, without invoking some more complex operation to deal with the fractional part?

The “Multiplication is Not Repeated Addition” Research:http://numberwarrior.wordpress.com/2009/05/22/the-multiplication-is-not-repeated-addition-research/

and

“Multiplication is Not Repeated Addition” Revisited:http://numberwarrior.wordpress.com/2010/02/26/multiplication-is-not-repeated-addition-revisited/

It was a rather half-hearted comment after all.

I used the phrase once before, with my wife, because of her discomfort with dealing with fractions / decimals / percentages. It helped her to see things needn't always be overly complex. Gave her the confidence for her to do store percentages in her head, rather then ask me all the time (what's 30 % off of 59.99?) , of course, I did help with a trick or two ;-).

She is quite pleased with her newly found superpower ;-)For any kind of arithmetic, even involving true real numbers, Goodhart is correct that it can all be reduced to addition and subtraction. My comment was meant in a more purely mathematical sense -- I don't thing that x

^{y}, where y is a potentially irrational or even complex number, can be implemented with just addition and subtraction....Unless...if you do a Taylor expansion, then you get an (infinite) sum of integer power terms, which does in fact reduce to multiplications and thence to additions. Okay, if you're willing to do an infinite number of additions, then Goodhart is correct in all cases.

In thermodynamic information theory, entropy turns out to be equal to the logarithm of the number of microstates available to a given macroscopic system.

In quantum chromodynamics (QCD, the current best theory of the strong interaction), many higher-order terms in interactions scale with ln(Λ

_{QCD}), the energy scale of the strong force.Mathematically, logs appear in many places because the integral of 1/x is ln(x).

Sorry, I just couldn't resist!

Now the constructive part

. I went digging through boxes, trying to find mine (I have 5-6), to no avail. Not sure I'd remember how to use them if I found them. :(

.

. For those of you that don't know what they are, check out the Wikipedia page on slide rules.

Somewherein a box, I have a real nice 10-12" Pickett that has more scales than any one person would ever need. Also have a couple of 8-10"ers that my Dad (BSEE) gave me. Plus several "novelty" ones (3-4", circular, &c). They are one of the things from The Good Ol' Days that I don't miss - give me a calculator any day.. Kind of amazing how they all but disappeared in just a few short years (1973-75, IIRC).

http://gwydir.demon.co.uk/jo/numbers/machine/log.htm

L

^{x}=32. To solve for x using logarithms, the equation can be written log_{2}32=x, which is read as "log base 2 of 32". log_{2}32 can then be solved to equal 5, so we know that 2^{5}=32.Whenever you see ln(x), that is the

natural logarithm...i.e., log_{e}x. This is used a lot when working with e (Euler's number, about 2.71828). Another special one is log(x), which is thecommon logarithm, and is the same as writing log_{10}x.Most scientific or graphing calculators cannot solve for any bases except e and 10. Fortunately, logarithms have the property that log

_{a}x=log(x)/log(a)=ln(x)/ln(a). So, to enter log_{2}32 into a calculator, you would have to enter log(32)/log(2), or ln(32)/ln(2).I hope I have helped, and am not just confusing you further.