So in my biology class, my professor wants us to use the Nernst equation without using calculators. I personally think this is stupid. However, I have no choice, so today, I tried coming up with approximations of the log function.
We start with loga(b) = n, and we want to find n.
The Attempt at a Solution
b = an
b = (1+(a-1))n
b ~ 1+n(a-1)
n ~ (b-1)/(a-1)
The problem with this approximation is that it's only good when a~b. I suppose I could decompose the log function using the rule log(ab) = log(a) + log(b), but this is somewhat tedious.
f(x) = 1 + xf'(x0)/1! + x2f''(x0)/2! + ...
Let a=e and b=1+x.
loge(1+x) = x - x2/2 + x3/3 - x4/4 + ... xn/n
This one is pretty bad. It's even worse than method 1, and the more terms I use, the worse it gets. I'm not sure what's going wrong here, but I think it's because xn grows much faster than n. Therefore, this approximation is only good for values close to 1, which is useless, since loge(1) = 0.
Furthermore, changing the base to loga(1+x) form, I'd have to divide by loga(e), which is another problem.
Any help will be much appreciated.