Saturday, August 18, 2007

Yes, logs again I'm afraid.

What I had wanted confirming in my other post was whether,

\displaystyle \frac{1}{2}\ln (1+x)^2 = \ln(1+x)

Straight after I'd posted the earlier post, it had been food time. I did ponder on logs for a while, but then became occupied with other things. As I sat playing Resistance Fall of Man*, I wondered again and convinced myself that I had done the right thing. I'm not very good at convincing myself it seems, but I hope the following is correct in convincing me.

We begin by introducing the variable z \in \mathbb{R}, and letting the problematic expression equal it:

(1)\begin{array}{cccc} \frac{1}{2}\ln (1+x)^2 &=& z\\ \ln(1+x) &=& z& \text{    Using `log rules'} \\ e^z &=& 1+x &\text{    A `log' thing}\\ \end{array}

That has been done using the rule which I was having a problem with. Now we do it without using that particular rule:

(2)\begin{array}{cccc} \frac{1}{2}\ln (1+x)^2 &=& z\\ \ln(1+x)^2 &=& 2z&  \\ e^{2z} &=& (1+x)^2 &\text{    A `log' thing}\\ (e^z)^2 &=& (1+x)^2\\ \end{array}

If I'm right in what I've done then if we 'square root' the final line in (2), then both (1) and (2) are equal. (I think this is dodgy maths, or I'm doing something rubbish, but I don't think its important to mention whether it is the positive or negative root). Actually, I may naively say that it doesn't matter, since both (1) and (2), if written in terms of x are, x= e^z -1 (regardless of the sign of the root and if I have not done anything stupid!)

There you have it folks - trivial things cause me the biggest problems. The problem tends to be the fact that I can't let go of questions like this. They're like a thorn in my side and no pain killers work! But it is always logs, regardless of it being integration or finding the limit of convergence. Somebody needs to go back to school.


*Any other Resistance Fall of Man players out there? I hadn't played for a couple of weeks, and upon recently playing, I discovered the clan I'd joined has disappeared! I had actually liked that clan, and am investigating what happened. :( (The clan leader seems to have vanished.) Hmmm, maybe it is about time I created my own clan! Well I have a name now at least - 'The Galois Group'. :D If you want to join let me know. ;) [There is this one player whose user name is Euler! I'll send him an invite.] Apart from when my brother is watching me play and muttering spells under his breath, I'm slowly getting back into the swing of things.

No comments: