Common and Natural logarithms: An Overview of Logs

Logarithms are an important concept in mathematics. The log of a number is the number of times it needs to multiply itself to get a given number. In this article, we have marked out very clearly what logarithms are and the difference between a ‘log’ and a ‘ln’. 

We have also included examples to make it easy for you to grasp the concepts. We have also listed out the relationship between common and natural logarithms and exponents and the properties of logs by the end of the article. 

What is a logarithm?

In very simple terms, the logarithm of a number denotes how many times it has to multiply itself to get another number. For example, we need to multiply 2 a certain number of times to get an even number. Let us take a random even number, 8. We know we have to multiply the number 2 three times to get 8. Here, 3 is the logarithm and the mathematical expression for such a calculation would be:

We know 2×2×2 = 8; 

is just another way of expressing the above equation.

The number that multiplies itself to produce another number is called ‘the base’. 2 is the ‘base’ in the example we used and the logarithm is 3. 

The expression can be said in three different ways:

• The logarithm of 8 with base 2 is (or is equal to) 3.
• Log base 2 of 8 is 3.
• The base-2 log of 8 is three.

It doesn’t matter which way you express it as long as you understand the concept. Just keep in mind that we are dealing with three numbers in a logarithm calculation—one, the number that gets multiplied by itself (the base), two, how many times the number is multiplied (logarithm) and three, the result of the multiplication or the number we want to get by multiplying the base a certain number of times (which is 8 in the above example).

Now, let’s look at what natural and common logarithms are. Both are widely used in mathematics. These two types of logarithms differ from each other in regard to the base they use.

Common logarithms:

Sometimes, we write a logarithm without mentioning its base. For example, log (100). Usually, this means that 10 should be taken as the base. Logarithms with 10 as the base are called common logarithms.

Common logarithms indicate how many times we have to multiply 10 to get the desired number.

For example, log (1000) is same as 

Since 10 ×10×10 = 1000, .

Natural logarithms:

Natural logarithms use base e for their calculations. ‘e’ is a base that is commonly used in mathematics. It stands for Euler’s number and has a value of 2.71828. A logarithm that takes e for its base is called a natural logarithm. Such a logarithm is represented by ‘ln’.

Natural logarithms or ‘ln’ shows how many times we have to use e in multiplication to get our desired number.

For example, ln (7.389) = log_e (7.389) ≈ 2

The logarithm here is approximately equal to 2 because 2.71828² ≈ 7.389.

Difference between log an ln

You must have understood by now that log and ln are different from each other and will give different values with the same number. This is because of the different base values they have. Still, it is possible to confuse the log and ln of a number when one is not being careful.

• The log is the common logarithm whose base is taken as 10, while ln is considered as the natural logarithm which has a base value of ‘e’ which is 2.71828.
• When the base is not mentioned with a log, it is always a common logarithm and 10 should be considered as the base.
• When ln is used, it is implied that e is the base, and therefore one should be careful not to read that as log and mistake it for a common logarithm.
• Calculators have both options for natural and common logarithms separately as ‘ln’ and ‘log’ respectively. 

Relationship between logarithms and exponents

Let us recall what exponents are. An exponent indicates the number of times a number is used in a multiplication. For example, a square of a number is an exponent that indicates the number gets multiplied by itself for once.

In the example, 2³, we know 2 is multiplied thrice and hence it is equal to 2 x 2 x 2. 

3 here is the exponent or the power of 2. Here, the number 2 can be understood as the base. A logarithm question is basically trying to find the exponent or power of a number that will give a certain number as an answer.

2^? = 8 is what a log question is trying to ask. As discussed above we express it this way,

To figure the R.H.S we just need to know how many times 2 gets multiplied to get the number 8.

Let us check your understanding of log and exponents with a sample question:

Q. Solve the value for x in log₂ x = 4

Solution: The logarithm function of the above question can be written as 2⁴ = y

We know, 2⁴ =2 x 2 x 2 x 2 which is 16.

Hence the value of y is 16.

Properties of logarithms

Logarithms have four basic properties or rules. The properties of logarithms follow from the fact that logs are derived from exponents.

1. Product rule: The multiplication of two logarithm values is equal to the sum of their individual logarithm values.

log_b(xy) = log_bx log_by.

1. Quotient rule: The division of two logs is equal to the difference between the individual logs.

               log_b(x/y) = log_bx – log_by.

1. Exponential or power rule of logarithm: The exponential rule of logarithm states that the log of an exponential number will be equal to exponent times the log of the base.

               log_b(xⁿ) = n log_bx.

1. When two logs are divided with the same base, the exponents would be subtracted.

log_bx = log_ax / log_ab.

Conclusion

The word ‘logarithm’ comes from the Greek word, logos, which has many meanings including ‘ratio or proportion’. They are important in many fields which require you to have a comprehension of advanced level mathematics. Engineers usually use the common logarithm while mathematicians use the natural log. It is important not to confuse one with the other and to have a strong foothold in your understanding of logarithms and exponentiation.