Logarithms
Put another log on the fire!
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It's my experience that logarithms confuse people. Let's look at the Big Word Book to see if we can start a blaze and get enlightened.
A logarithm is a power
to which a base [like 10] must be raised to produce a given number. If nx = a, the logarithm of a, with n as the base, is x. Symbolically this is logna = x. For example, 103 = 1,000; therefore log101000 = 3.This is probably still confusing, so let's break it down a bit.
First, the word power is the same as exponent. Thinking whole numbers, this is the number of times a base has to be multiplied by itself. So 32 shows the power 2 and means you just multiply 3 by itself, 3x3. The power could be any number. The base, in turn, is the number being multiplied by itself. It can be any number greater than zero but not equal to 1. (The number 1 raised to any power is still 1, so it's useless as a logarithm.)
Since practically any number can be a base, we could have many types of logarithms -- like having different languages. But only a few are considered important enough to develop. One is the common, or base 10, system. A second is the so-called natural system, which uses a strange number as its base, called e, which we don't need to get into here. And a third has become more interesting in this age of digital computing; this is the system having 2 as its base. (You will appreciate that, say, log28 = 3 is a restatement of 23 = 8.)
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According to my mathematical handbook, which I used a hundred years ago in college, the logarithm x = logca to the base c>0 (c¹ 0) of the number a > 0 may be defined as the solution of the equation:
cx = a
It says c raised to the power x is a. (In base 10 logarithms, this says, for instance, 102 = 100.) Solving the equation means finding the value of x that satisfies the equation. However, since the equation has three variables, you can't find x without specifying values for both c and a. You'd likely start with the need to give the log of a. So you'd be given a and select the base, c, in whose terms you wish to give the answer. In other words, you would choose a type of logarithm for your problem.
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Since the same equation determines all logarithms, there should be equivalents among them. Suppose we express cx = a, in the following two forms, using base 10 and base 2 logarithms:
10x = a
2y = a
where x = log10a and y = log2a. Since two things equal to the same thing are equal to each other, we write:
10x = 2y
or
10log10a = 2log2a
You can make similar comparisons using base 10 with base e, or base 2 with base e.