(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

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.

2. Relevant equations

We start with log_{a}(b) = n, and we want to find n.

3. The attempt at a solution

Method 1:

b = a^{n}

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.

Method 2:

f(x) = 1 + xf'(x_{0})/1! + x^{2}f''(x_{0})/2! + ...

Let a=e and b=1+x.

log_{e}(1+x) = x - x^{2}/2 + x^{3}/3 - x^{4}/4 + ... x^{n}/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 x^{n}grows much faster than n. Therefore, this approximation is only good for values close to 1, which is useless, since log_{e}(1) = 0.

Furthermore, changing the base to log_{a}(1+x) form, I'd have to divide by log_{a}(e), which is another problem.

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Any help will be much appreciated.

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# Homework Help: Good Approximation to the Log Function

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