- #1

- 209

- 0

## Homework Statement

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.

## Homework Equations

We start with log

_{a}(b) = n, and we want to find n.

## 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.

------

Any help will be much appreciated.

Last edited: