# Definitions

## Exponents

Exponents are just a shorthand for long multiplication problems. Instead of writing out *x* · *x* · *x* · *x* · *x* · *x* we would write *x*^{6}, 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 5^{4}.

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(10^{3}) = 3

log(100) = log(10^{2}) = 2

log(10) = log(10^{1}) = 1

log(1) = log(10^{0}) = 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 10^{1.079} = 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 log_{2}*x* is "the number to which you have to raise 2 to get *x*". Revisiting our previous examples

log_{2}(8) = log_{2}(2^{3}) = 3

log_{2}(4) = log_{2}(2^{2}) = 2

log_{2}(2) = log_{2}(2^{1}) = 1

log_{2}(1) = log_{2}(2^{0}) = 0

log_{2}(.5) = log_{2}(2^{-1}) = -1

So what about log_{2}5.4? 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:

log_{n}x = |
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

log_{2}5.4 = |
log(5.4) | = | 0.7323 | = 2.432 |

log(2) | 0.3010 |

You can confirm with a scientific calculator that 2^{2.423} = 5.4.