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Definitions

Exponents

Exponents are just a shorthand for long multiplication problems. Instead of writing out x · x · x · x · x · x we would write x6, i.e. the variable and the number of times we wanted it to be multiplied by itself. The superscript part of the expression is called the exponent; the lower part, the x in our example, is called the base. Of course, we can do this with numbers as well as variables so, for example, 5 · 5 · 5 · 5 would be 54.

Our definnition is based on multiplication so all of our exponents intially are natural numbers, 1, 2, 3, etc. The procedure has natural extensions to negative values and even fractions.

Logarithms

Logarithms are a little trickier to define using English. The logarithm of a number is the number to which you'd have to raise ten to get the original number. The multiple uses of "a number" make the definition a little hard to follow. It's usually a little easier to look at some examples.

log(1000) = log(103) = 3
log(100) = log(102) = 2
log(10) = log(101) = 1
log(1) = log(100) = 0
log(.1) = log(10-1) = -1

So how about numbers that aren't a power of 10? For those you usually need to get out your calculator. The thinking goes like this:

log(12) = 1.079 because 101.079 = 12
log(.12) = -0.9208 because 10-.09208 = .12

Why does the logarithm always represent a power of 10? It doesn't, necessarily. If no other base is specified then the assumption is that it's 10 but you can specify other bases by using a subscript. For example log2x is "the number to which you have to raise 2 to get x". Revisiting our previous examples

log2(8) = log2(23) = 3
log2(4) = log2(22) = 2
log2(2) = log2(21) = 1
log2(1) = log2(20) = 0
log2(.5) = log2(2-1) = -1

So what about log25.4? Or some other number that isn't a power of 2? Here you can run into problems since calculators don't have keys for every possible base. To calculate these values you can use a simple formula:

lognx log(x)
log(n)

Applying this to my question at the beginning of the first paragraph, and using a calculator to get the base 10 logarithm values, gives us

log25.4 =  log(5.4)  =  0.7323  = 2.432
log(2) 0.3010

You can confirm with a scientific calculator that 22.423 = 5.4.