What does Heron's Algorithm have to do with the construction of logarithmic tables

Question

i need a little help answering this question, what does Heron's Algorithm have to do with the construction of logarithmic tables.

I know that Heron's algorithm is used for finding square roots, but how does that come in to play with the construction of logarithmic tables?

Any help is appreciated

Answer 1

An algorithm for finding square roots is useful for computing logarithm tables because it can be used to find a new point in your logarithm function between two which you have computed already.

Initially you can compute some points in your logarithm table by multiplying the base of your logarithm over and over like so:

• $\log_b(1) = 0$
• $\log_b(b) = 1$
• $\log_b(b^2) = 2$
• $\cdots$

If you then consider two existing point in your logarithm table, you can compute a point between them by using the square root. Assume you have

• $\log_b(x_1) = y_1$
• $\log_b(x_2) = y_2$

You can compute an entry between the two as

$$\log_b(\sqrt{x_1 x_2}) = \frac{x_1 + x_2}2$$

This way you can fill in the gaps in your logarithm table to any density you like.