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what exactly is a log? could you define what logs are used for and why? what does it mean to have something log based 10?

A logarithm of a number N (to remind us it is the number), is the power (exponent), lets call it L (to remind us it is called the logarithm) that a number, let's call it B (to remind us it is called the base) must be raised to equal N. So, symbolically this can be written:

B^L =N so the log (N) to the base (B) is L. A couple of examples makes
it clearer than trying to give a this general symbolic expression, which
is really clumsier than the concept itself.

1000 = (10)^3 so 3 is the logarithm of 1000 to the base 10. The symbol "^" means raised to the power of.
This is written: log10(1000)=3

16 = (2)^4 so 4 is the logarithm of 16 to the base 2.
This is written: log2(16) = 4

Base 2 logarithms are used in computers a lot because computers ultimately come down to two choices 1 or 0. So you may have seen the number 1024 crop up in a lot of stuff about computers. It is: 1024 = (2)^10 or 10 = log2(1024)

See! Taking a logarithm is just the opposite, the inverse function, of taking a power. It is the "un-power" of a number.

The base number can be any number that is convenient as long as it is clear to the reader what it is. In fact, a base that keeps turning up in a lot of math, physics, and chemistry is the base number, e , where e = 2.718281828... is a number whose decimal expression never repeats itself, although it might not look so by only looking at the first 9 digits, but trust me, it just starts wandering off into "number-number land". So we can write:

loge (6) = 1.791759469... Because this base is very common, although it seems a little wierd, it is given a special symbol "ln" . So ln(6) = 1.791759469...

Vince Calder

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