1. What are logarithms used for?

Before the days of calculators, logarithms were used to do long multiplication and division. Tables of the logarithms to base10 and base e were published and students would take them into examinations, much the way calculators are used today.

Logs are also used to solve some types of index equations.

2. Why are the laws of logarithms so similar to the laws of indices.

Because a logarithm is an index, to base 10 or e.

3. What is the difference between the  button and the  button on most calculators?

The  button gives logs to base 10.

The  button gives logs to base e.