## 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.

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How it Works »New Instructable »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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Mar 1, 2009

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active| newest | oldest_{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/