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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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Cameron's response below gives you a good definition of a logarithm: logb(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: bx×by = bx+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 bx/by = bx-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.

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.

That is true of real numbers. In the complex plane, you can rotate a number without adding or subtracting anything; and don't get me started about quarterions.

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 (n3 = 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?

If you want enrich your  "multiplication is equivalent to repeated addition" comment, go to:

The “Multiplication is Not Repeated Addition” Research:

and

“Multiplication is Not Repeated Addition” Revisited:

Sorry, I hadn't meant to ruffle feathers :-)

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 ;-)

You're both right, in an operational since. Since computers only recognize fixed-precision bit patterns (whether integer or "floating point"), you can reduce all operations to just adds. Extending Nacho's comment, all you really need is NAND (not-and). Every other operation can be implemented as cascades of those.

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 xy, 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.

For completeness, I should explain my somehwat cryptic opening comment, "not much unless you are a physicist specializing in either thermodynamics or QCD."

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(&Lambda;QCD), the energy scale of the strong force.

Mathematically, logs appear in many places because the integral of 1/x is ln(x).

Simply put, a logarithm is an exponent-related thing which, after a while, makes sense. Basically, you use the base 10. You get a random number, maybe 1000, then find out how many times you have to multiply the base by in order to find the logarithm. In this case, the log of 1000 is 3. And yes, other digits also work, but you better find a slide rule because that's the nearest you can get to calculating with logs (aside from digital computers).

I haven't used a slide rule in about 30 years :-) My dad got me a really nice one (12 inches long, multiple scales including a reversible center slide) when I was 10. Calculators are more convenience and provide better precision when needed; slide rules accumulate round-off errors when performing long computations.

. Somewhere in 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).

1973-1975 is probably right. That's when TI took their excellent military cryptographic techology (integrated circuits), combined it with these new-fangled light-emitting diodes, and started to make inexpensive consumer products with them. The rest, to quote the irritating cliché, is history.

> inexpensive . Not hardly! IIRC, the 4-function TI's were selling for around \$200. That's \$200 in mid-'70s dollars (?about \$500 in current dollars?) for add/subtract/multiply/divide. No square root. No log. Just +, -, x, and *. No trig functions. Just +, -, x, and *. \$200. . And battery life was awful. . Does anyone even make a Plain Jane, 4-function calculator anymore? Even the keychain models seem to all have at least one memory register and a sq root key.

It is a way of solving for an exponent. Lets say that you know that 2x=32. To solve for x using logarithms, the equation can be written log232=x, which is read as "log base 2 of 32". log232 can then be solved to equal 5, so we know that 25=32.

Whenever you see ln(x), that is the natural logarithm...i.e., logex. This is used a lot when working with e (Euler's number, about 2.71828). Another special one is log(x), which is the common logarithm, and is the same as writing log10x.

Most scientific or graphing calculators cannot solve for any bases except e and 10. Fortunately, logarithms have the property that logax=log(x)/log(a)=ln(x)/ln(a). So, to enter log232 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.