Is there a simple way for natural logs be calculated by hand?

Question

Why are natural logs not calculated by hand often? Is it too difficult to get a accurate answer without a calculator?

Answer 1

Remember how your were taught how to add, subtract, multiply and divide when you were a young child? These are very easy to do by hand. It's very hard to do anything else, e.g. square roots, cube roots, trigonometric functions and logarithms are very hard to do by hand.

It is possible to do, but you need to break them up into the four basic operations. It's possible to write logs as addition and multiplication, as follows:

A series expansion is the best way to calculate approximate values.

For example, for some values of $x$, the Taylor Series expansion is $$\ln\left(\frac{1}{1-x}\right) = x + \frac{1}{2}x^2+\frac{1}{3}x^3+\frac{1}{4}x^4+\cdots+\frac{1}{k}x^k+\cdots$$

If you want to approximate $\ln 2$, then substitute $x=\frac{1}{2}$:

$$\ln 2 \ \ \approx \ \ \frac{1}{2} \ \ + \ \ \frac{1}{2}\cdot \frac{1}{4} \ \ + \ \ \frac{1}{3}\cdot\frac{1}{8} \ \ + \ \ \frac{1}{4}\cdot \frac{1}{16} \ \ + \ \ \frac{1}{k}\cdot\frac{1}{2^k} $$

The more terms you add, the better the approximation. For example

When $k=5$, we get $\ln 2 \approx 0.68854167$ which is off by $0.7\%$

When $k=10$, we get $\ln 2 \approx 0.69306486$ which is off by $0.01\%$

In reality, $\ln 2 = 0.69314718$ to 8 significant figures.