Math
Logarithms

Refresh your memory of exponents if you need to, but in the equation 3² = 9, the little 2 is the exponent of 3 (the number of times it is multiplied by itself), and in 10² = 100, the little 2 is the exponent of 10. What Napier did, more or less, was to say that a different way, namely: "the logarithm of 100 is 2." That is, the logarithm is the power to which 10 must be raised to give the number in question. We know that 10¹ = 10, so that the logarithm of 10 is 1. To sum up so far:

log 1 = 10
log 2 = 100

It follows that the numbers between 10 and 100 must correspond to logarithms between 1 and 2, numbers such as 1.5. How, you ask, can a number be multiplied by itself 1.5 times? It can't, but if we go ahead anyway, we can easily determine what number corresponds to log 1.5. First, remember that when two powers of the same base are multiplied, their exponents add: (x )(x ) = x . then, thinking of base 10, we would have

(10*1)(10*0.5) = 10*1.5

Now, 10*0.5 is the same as Ö10. By trial and error (or by desk calculator) we know that Ö10 = 3.16278. Then we can pair the above equation with this one:

(10*1)(10*0.5) = 10*1.5
(10) ( Ö10) = (10)(3.1628) = 31.6278

From which it follows (to repeat our above table) that:

log 1 = 10.0000
log 1.5 = 31.6278
log 2 = 100.0000

And by such processes, we can calculate our own table of logarithms.

We remember that 10*1/2 = 10*0.5 is another way of writing the square root of 10, or Ö10 (since Ö10 x Ö10 = 10). We can get the square root of 10 by thinking: Hmm, it has to be a little more than 3, the square root of 9. . . Or by using our desk calculator, which gives 3.1623).

Exponents add, so (10)(Ö10) = 10*1 x 10*0.5 = 10*1.5 = 31.623. This is a necessary consequence of things we already knew. So the idea of 10 to the one-and-a-half power not only has numerical meaning, we can very easily work out its numerical value.

the tree? Or one and aThe exponent is the little number to the upper right of a letter or numeral, like the little 2 in the expression x². It means that the number in question is to multiplied by itself that many times, thus

x² = (x)(x)
x³ = (x)(x)(x)

and so on. x² is conventionally read as "x squared." More generally, an exponent means, and can be pronounced, "x raised to the [whatever] power."

Roots

One may ask the opposite question: what is the number which, when multiplied by itself [however many] times, equals x? For x² the answer is "the square root of x," symbolized by Öx ("radical x" or "root x"). For greater precision, this might be written ²Öx, but we typically do not bother; all roots are assumed to be square roots unless written otherwise. If another index, say n, is written before the radical sign, the resulting expression is pronounced "the nth root of x."

Operations

Briefly, when powers of the same base are multiplied, their indices (exponents) are added, thus:

(x¹) (x²) = x³

Substitute 2 for x in these equations to demonstrate their truth.

It follows that when powers of the same base are divided, their indices are subtracted, thus

x³ /x² = x¹

It follows that:

x³ /x³ = x¹ = 1

which is to say, anything to the zero power (anything divided by itself) equals 1.

Indices of different base (as, x² times y²) cannot be multiplied; their product can only be indicated as x²y² (and so on). Powers with the same base and the same exponent (and only those) may be added, thus:

x² + x² = 2x²

When evaluating an expression like 2x², we always do the exponentiation first, and only then the multiplication. Thus, if x here equals 3, we have first

x² = 9

and then, substutiting in the expression 2x², we have

(2)(9) = 18

or in one step, 2x² = 18. The big "2" out in front is called a coefficient. Mistakes with exponents are usually mistakes of order in evaluating expressions like 2x², or mistakes in multiplication. If the above models are followed, there should never be any trouble.

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