# How are two logarithms multiplied together

## Powers, roots and logarithms¶

### Calculation rules for logarithms¶

In addition to taking the root, taking the logarithm is a second possibility, a power to find that a certain result supplies. While taking the root of the (root) exponent is given and the base matching the value of the power is sought, taking the logarithm helps to establish the basis on a given basis matching exponent to find. The question is thus: To solve this math problem, what is known as the logarithm of to the base be determined.

Definition:

The logarithm is the number with which the base Must be raised to the power of the result to obtain. The following applies: For example, the following applies as can be verified by inserting it into the left part of the equivalence equation above, as well as , there corresponds exactly to the number with which the base Must be raised to the power of the result to obtain.

A simple calculation of a logarithm “by hand” is generally only possible in rare cases. In the past, tables of values ​​for logarithms were printed in textbooks and formulas; in the meantime, pocket calculators or computer programs with corresponding functions have significantly simplified the calculation of logarithms and ultimately made tables of values ​​superfluous.

In practice, logarithms in particular are the basis ("Decadic" logarithms, symbol: ), to the base ("Natural" logarithms, symbol: ) and to the base ("Binary" or dual "logarithms, characters or ) is important.  To convert a logarithm to another base, the following formula can be used:

(10)¶ For example, the above formula makes it possible to use a decadic logarithm into a binary logarithm to convert by passing this through Splits.

Sums and differences of logarithms

Logarithms with the same base can be added or subtracted. The result of a logarithm addition is a logarithm with the same base, the argument of which is equal to the product of the arguments of both logarithms to be added:

(11)¶ Accordingly, the result of a logarithm subtraction is a logarithm with the same base, the argument of which is equal to the quotient of the arguments of both logarithms to be subtracted:

(12)¶ Becomes a logarithm with a constant factor multiplied, this corresponds to one -Fold addition of the logarithm with itself. In this case the result corresponds to a logarithm with the same base , whose argument -fold must be multiplied by itself:

(13)¶ Logarithm equations are dealt with in more detail in the context of elementary algebra, and logarithm functions in the analysis chapter.

Remarks: