1. What are logarithms used for?

In the past, before calculators were as cheap as they are now and before they were invented, logarithms were used to carry out long multiplication and division. Books of tables were used in the calculations which turned the multiplication and division into adding and subtractions by using the properties of logarithms.

Logarithms are also used for solving exponential equations and in mathematical modelling.

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

The  button is for logarithms to base 10, that is log10.

The  button is for logarithms to base e, that is log e.