# AS & A-Level Maths: Where the Power and Log Laws Come From

You will need to know the power laws and log laws for AS and A-level maths. The power laws and log laws may be old friends, but do you know where they come from? Imagine you have seven £2 coins. If you wanted to write down how much money you had, you could write £2+ £2+ £2+ £2+ £2+ £2+ £2, but you’d never dream of doing that (for one thing, you’d probably lose count); you’d almost automatically write 7×£2 or £2×7.

In a similar way, if you multiply things together repeatedly, you could write £500× 1.01× 1.01× 1.01 more efficiently as £500×1.01³. In both cases, the notation clarifies what’s going on and, critically, how much of it needs to happen.

There’s also some kind of a pecking order emerging: adding turns into multiplication, which turns into . . . let’s call it *powering*. (This, it turns out, is where the horrors of BODMAS come from; the rule is simply that you do the most general operation first, unless brackets or groupings tell you otherwise.)

## So, where do the power laws come from?

Imagine you need to work out (3×3×3×3)× (3×3×3), which you’d happily turn into 3^{4}× 3³. You could rewrite it as a list of seven 3s multiplied together or as 3^{7}. If you had *a* 3s in the first list and *b* 3s in the second, you’d turn them into a list of (*a*+*b*) 3s. More generally still, this leads to the rule (*x** ^{a}*)(

*x*

*)=*

^{b}*x*

^{a}^{+}

*.*

^{b}Divide both sides by *x*^{b}*,* and you get *x** ^{a}*= (

*x*

^{a}^{+}

*)/*

^{b}*x*

^{b}*;*if you divide the product of (

*a*+

*b*)

*x*s by the product of

*b*

*x*s,

*b*of them cancel out, leaving you with the product of

*a*

*x*s. You can rewrite this as

*x*

^{m}*/x*

^{n}

*= x*

^{m}

^{–}*– so the first couple of power laws hold water.*

^{n}Another consequence of the first power law is that if you multiply a power by itself several times – as in (*x** ^{a}*)

^{3}

^{ }– you can write it as (

*x*

*)(*

^{a}*x*

*)(*

^{a}*x*

*)=*

^{a}*x*

^{3}

*. In general, multiplying*

^{a}*x*

*by itself*

^{a}*b*times gives you

*a list of*

*b*(

*x*

*)s, which you can turn into*

^{a}*x*

*.*

^{ab}Roots aren’t quite as easy to see, but think about the example from the preceding paragraph: (*x** ^{a}*)(

*x*

*)(*

^{a}*x*

*)=*

^{a}*x*

^{3}

*, which you can rewrite as (*

^{a }*x*

^{b}^{/3})(

*x*

^{b}^{/3})(

*x*

^{b}^{/3}) =

*x*

^{b}*;*it’s now clear that

*x*

^{b}^{/3}is the cube root of

*x*

^{b}*.*You can extend this idea to see that

*x*

^{1/}

^{k}*is the*

*k*th root of

*x*.

## And the log laws?

The remaining log laws can be worked out in pretty much the same way: each of the log laws corresponds exactly to a power law!