## Logarithms Theory

# Logarithms

**Definition of Logarithms:** Be a positive real number non-zero and different to 1, and A another positive and non-zero number. It is called the logarithm of the number A in the base a, the exponent x to the base must be raised to obtain that number.

To indicate that x is the logarithm of the number A in the base a, it is written:

Which it is read as: “x is equal to the logarithm of A at the base a.”

Thus:

This equivalence allows us to get from the logarithmic notation to the potential (or exponential) notation and reciprocally.

__Decimals and neperian logarithms__

From the infinity of bases that we can choose, there are two with which are those that are used in practice:

Base a= 10 -> they are called logarithms or common logarithms those that are based on the number 10. Since they are used in practice it is usually unwritten the base; so that:

- Base a=e -> they are called natural or hyperbolic logarithms those that are base on the number e. They are written as follows:

**Properties of logarithms:** Managing logarithms requires good understanding of their properties. It cannot be overemphasised, especially in the properties relating the logarithms with arithmetic operations and the use is essential and indispensable in the logarithmic calculation.

**Changing Bases:** Given the affinity of bases that exist (all positive real number different to 1), it is impossible to have logarithms tables in all these bases or calculators that give us directly the logarithm of a number.

Logarithms of a number A in the base a and in the base b are related as follows:

This formula allows us to know the logarithms in base b knowing the base a ones.

**Pass from decimals logarithms to neperian logarithms and vice versa:** Applying the following formula we can get a logarithm from the other one:

#
You can download the App **BioProfe READER** to practice this theory with **self-corrected exercises**.