Basics of Logarithm
What is Logarithm?
Logarithms are related to the concept of exponents/powers/indices. In fact, they are another way of writing exponents/indices/powers.
We write logarithm as log in short. It is represented as
We read
For example,
Relationship between exponents and logarithms
In general terms,
We can represent the relationship between b, a and p using either equation. While changing one form of equation into another, remember that base of the logarithm is the same as the base of the exponent expression.
For example, we know that
In other words, we can say that we can get 8 from 2, if we multiply it 3 times, i.e.
Let’s see one more example,
We basically need to find out, “By what power should we raise 3 to get 27?”, or “How many times should we multiply 3 by itself to get 27?” We can see that 3 × 3 × 3 = 27. So, we need to multiply 3 three times by itself. So,
Some restrictions on the values of log variables
In a logarithm, say
Restriction on value of base of log
The base of log must be positive too, but not equal to 1.
Restriction on value of argument of log
The argument of log must be positive. Logarithms of negative numbers are not defined.
No restrictions on the value of a log
If
For example,
Special Logarithms
The base of logarithm can attain any value (except 1). But in most of the questions you encounter, you will come across two base values more often than others.
These are 10 and e. These two values are so common that most of the scientific calculators have dedicated buttons for them.
Common Logarithm
When the base of a log is 10, then we call it a common logarithm. For example,
We can use it to find the number of digits in any number. We know that,
For example,
So, if we know the value of the common log of a number, then we can tell how many digits it has before the decimal. Log of
For example,
Natural Logarithm
When the base of a log is e, then we call it a natural logarithm. For example,
Properties of Logarithm
Property 1
For example,
So,
Also,