Definition and Properties of Logarithms - Revision Notes

In these revision notes for Definition and Properties of Logarithms, we cover the following key points:

• What is a logarithm? How are they related to exponentials?
• What are the components of a logarithm?
• How did the history of logarithm begin?
• What are the properties of logarithm?
• How are the properties of logarithm combined to calculate the value of logarithmic expressions?

Definition and Properties of Logarithms Revision Notes

The logarithm represents another way of expressing the exponential form of a number, where the variable to be found is the exponent, not the power.

Given the general form of an exponential expression

where a is called the 'base', b is called the 'exponent' (or 'index') and c is the result of the operation called 'power', then the corresponding logarithmic form of the same expression is written as

The terms included in a logarithm are as follows:

The term 'a' is still called the base, like in the corresponding exponential form;

The term 'b' is called the logarithm, unlike in the corresponding exponential form, where it is called the 'exponent'.

The term 'c' is called the 'argument', unlike in the corresponding exponential for, where it was called the 'power'.

We read this new expression as "the logarithm with base a of c is b", or simply "log base a of c is b". The logarithmic form of a number or expression is used when the base a and the power b are known and we want to express the base b in terms of them,

If the base of a logarithm is not written, it is taken as 10. This means the notations log₁₀ x and log x are equivalent.

The invention of logarithms was cued by the need to find a connection between arithmetic and geometric sequences.

Logarithms have a number of properties that derive from their relationship with exponents. They are:

1. Product property rule

This property states that:

"The logarithm of a product is equal to the sum of the individual logarithms."

In symbols, the product rule of logarithms is written as

2. Quotient property rule

This property states that:

"The logarithm of a quotient is equal to the difference of the individual logarithms."

In symbols, the quotient rule of logarithms is written as

3. Power property rule

This rule says:

"The logarithm of a number raised in a given power is equal to the product of that power and the logarithm of the number itself."

In symbols, the power rule of logarithms is written as

4. Log of the same number as base property rule

If the base and argument of a logarithm are equal, then the following property is true:

This property is called the "log of the same number as base" rule. It is true because a¹ = a.

5. Change of base property rule

Sometimes, it is more suitable to change the base of a logarithm to make the operations easier. The change of base rule says:

It is possible to divide the logarithm of the old argument by the logarithm of the old base by expressing these new logarithms at the same new base; the result obtained does not change.

In symbols, the change of base rule of logarithms is written as

6. Equality property rule

This rule says that if two equal numbers written in the logarithmic form have the same base, then their arguments are also equal.

In symbols, we write this property as

7. Number raised to log property rule

Sometimes, the exponent of a number may contain a logarithm instead of an ordinary number. The number raised to log property rule says that

If a number is raised to a logarithmic power, where the base of logarithm is the same as the base of the given number, then the value of the expression is equal to the value of the logarithm argument.

In symbols, we write this property as

8. Exponential base of logarithm property rule

Sometimes, the base of logarithm is written in the exponential form. In these cases, the following property is applied:

When the base of a logarithm is expressed in the exponential form, its exponent can be written in the denominator of the fraction preceding the logarithm.

In symbols, we write this property as follows

9. Rational base or argument property rule

This property says:

If the base or argument of a logarithm (or both) are expressed as fractions, we can write them as rational powers and then continue using any of the previous properties of logarithm.

In symbols, we write it as

10. Rational base or argument property rule

This property says:

If the base or argument of a logarithm (or both) are expressed as fractions, we can write them as rational powers and then continue using any of the previous properties of logarithm.

In symbols, we write it as

All the above properties of logarithms are used to make the operations and calculations easier. In most cases, an exercise may require to combine more than one of the above properties in order to obtain the simplest form or the result of the expression.