# COMMUNITY : FORUMS : BURNING QUESTIONS

## 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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kelseymh8 years ago
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.
8 years ago
then you can look up the logarithms

hee hee, log tables....
8 years ago
Yeah, yeah, I should have used the past tense. I'm pretty sure even the CRC Handbook has dropped their log tables by now.
8 years ago
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.
8 years ago
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?
6 years ago
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: