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 34× 3³. You could rewrite it as a list of seven 3s multiplied together or as 37. 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 (xa)(xb)= xa+b.
Divide both sides by xb, and you get xa= (xa+b)/xb; if you divide the product of (a+b) xs by the product of b xs, b of them cancel out, leaving you with the product of a xs. You can rewrite this as xm/xn = xm–n – so the first couple of power laws hold water.
Another consequence of the first power law is that if you multiply a power by itself several times – as in (xa)3 – you can write it as (xa)(xa)(xa)= x3a . In general, multiplying xa by itself b times gives you a list of b (xa)s, which you can turn into xab.
Roots aren’t quite as easy to see, but think about the example from the preceding paragraph: (xa)(xa)(xa)= x3a , which you can rewrite as (xb/3)(xb/3)(xb/3) = xb; it’s now clear that xb/3 is the cube root of xb. You can extend this idea to see that x1/k is the kth 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!