Let us assume that, a logarithm is said to be an exponent when a base 2 is given with an exponent 3, then how will you calculate 23? As you know  23= 2 * 2* 2= 8,  suppose a base 2 is given with a power 8, and you are asked to find the exponent  2? = 8, hence the exponent that produces the value is called as Logarithm. Here 23= 8 so the exponent 3 is nothing but the logarithm of 8 with a base value of 2. In a logarithmic way, we write it as 3 = log28. In the same way 104 = 10,000, it can be written as log1010,000 = 4. In a general way, the logarithmic function can be written or denoted as

Logb x = n  or   bn = x

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John Napier introduced the concept of Logarithms in the 17th century. Later it was used by many scientists, navigators, engineers, etc for performing various calculations which made it simple. In simple words, Logarithms are the inverse process of the exponentiation. In this article, we are going to have a look at the definition, properties, and examples of logarithm in detail.

Logarithm Definition

A logarithm is defined as the power to which number must be raised to get some other values. It is the most convenient way to express large numbers. A logarithm has various important properties that prove multiplication and division of logarithms can also be written in the form of logarithm of addition and subtraction.

“The logarithm of a positive real number a with respect to base b, a positive real number not equal to 1[nb 1], is the exponent by which b must be raised to yield a”.

i.e by= a and it is read as “the logarithm of a to base b.”

In other words, the logarithm gives the answer to the question “How many times a number is multiplied to get the other number?”. 

For example, how many 3’s are multiplied to get the answer 27?

If we multiply 3 for 3 times, we get the answer 27. 

Therefore, the logarithm is 3.

The logarithm form is written as follows:

Log3 (27) = 3 ….(1)

Therefore, the base 3 logarithm of 27 is 3.

The above logarithm form can also be written as:

3x3x3 = 27

33 = 27 …..(2)

Thus, the equations (1) and (2) both represent the same meaning.

Logarithm Types

In most cases, we always deal with two different types of logarithms, namely

  • Common Logarithm
  • Natural Logarithm

Common Logarithm

The common logarithm is also called the base 10 logarithm. It is represented as log10 or simply log. For example, the common logarithm of 1000 is written as a log (1000). The common logarithm defines how many times we have to multiply the number 10, to get the required output.

For example, log (100) = 2

If we multiply the number 10 twice, we get the result 100.

Natural Logarithm

The natural logarithm is called the base e logarithm. The natural logarithm is represented as ln or loge. Here, “e” represents the Euler’s constant which is approximately equal to 2.71828. For example, the natural logarithm of 78 is written as ln 78. The natural logarithm defines how many we have to multiply “e” to get the required output.

For example, ln (78) = 4.357.

Thus, the base e logarithm of 78 is equal to 4.357.

Logarithm Properties

There are four basic rules of logarithms as given below:-

  1. Logb (mn)= logb m + logb n. In this rule, the multiplication of two logarithmic values is equal to the addition of their individual logarithms.

          For example- log3 ( 2y ) = log3 (2) + log3 (y)

  1. Logb (m/n)= logb m – logb This is called as division rule. Here the division of two logarithmic values is equal to the difference of each logarithm.

         For example, log3 ( 2/ y ) = log3 (2) -log3 (y)

  1. Logb (mn) = n logb m

         This is the exponential rule of logarithms. The logarithm of m with a rational exponent is equal to the exponent times its logarithm.

  1.  Logb m = loga m/ loga b    

Also check:

Logarithms Examples

Question 1: Solve log 2 (64) =?


since 26= 2 ××××× 2 = 64, 6 is the exponent value and log 2 (64)= 6.

Question 2: What is the value of log10(100)?


In this case, 10 2 yields you 100. So, 2 is the exponent value, and the value of log10(100)= 2

Question 3: Use of the property of logarithms, solve for the value of x for log3 x= log3 4+ log3 7


By the addition rule, log3 4+ log3 7= log 3 (4 * 7 )

Log 3 ( 28 ). Thus, x= 28.

Question 4: Solve for x in log 2 x = 5


This logarithmic function can be written In the exponential form as 2 5 = x

Therefore, 2 5= 2 ×××× 2 = 32, X= 32.

Logarithms Applications

Logarithms are nowadays widely used in the field of science and technology. We can even find logarithmic calculators which have made our calculations much easier. These find its applications in surveying and celestial navigation purposes. They are also used in calculations such as measuring the loudness (decibels), the intensity of the earthquake regarding Richter scale, in radioactive decay, to find the acidity (pH= -log10[H+]), etc.

Frequently Asked Questions on Logarithm

What is meant by logarithm?

A logarithm is defined as the power to which a number must be raised to get some other values. In other words, it gives the answer to the question “How many times a number is multiplied to get the other number?”. The logarithm of a number is expressed as
logb x = y

What are the two different types of a logarithm?

The two most common types of logarithms are:
Common Logarithm (or) Base 10 Logarithm
Natural Logarithm (or) Base e Logarithm

Mention any two properties of the logarithm?

The two important properties of logarithm are:
Logb (mn) = logb m+ logb n
Logb (m/n) = logb m – logb n

What is the logarithm of 0?

The logarithm of 0 is undefined. Because, we never get the value 0, by raising any value to the power of anything else.

What is the logarithm of 10?

The logarithm of 10 is 1. (i.e.,) log10 10 = 1. Hence, the base 10 logarithm of 10 is 1.

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Test your knowledge Logarithms


  1. log3^2*log2^81=4


    asa sir i want solution of this

    1. 1) log32*log281=4
      log32 = 0.63
      log281 = 6.34
      ⇒ 0.63*6.34 = 3.99= 4

      2) log53 * log325 = 2
      log53 = 0.68
      log325 = 2.93
      ⇒ 0.68*2.93= 1.99=2

    2. log3^2×log2^81=4
      1/log2^3×log2^81=4 (1/log2^3=log3^2)
      4log2^3/log2^3=4 (log2^81=4log2^3)
      4 = 4

      1/log3^5×log3^25=2 (1/log3^5=log5^3)
      2log3^5/log3^5=2 (log3^25=2log3^5)
      2 = 2

  2. a power (log base b (log base b n)/log a base b)

    1. Let, Log1024 base4 = X
      4^X = 1024 (by log formula)
      4^5 = 1024
      Therefore, X=5
      Log1024 base4 = 5

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