Value of Log 1

• In Mathematics, most of the researchers used logarithms to transform multiplication and division problems into addition and subtraction problems before the concept of calculus came into the picture.

• Logarithms are continuously used in Mathematics and Science as both subjects contend with large numbers.

•  In this article, we will discuss the log 1 value (log 1 is equal to zero) and the method to derive the value of log 1 through common logarithm functions and natural logarithm functions.

A logarithm is an inverse expression of an exponent. For example, n = b^x , where n is a real positive number. And x is the exponent number. Then, the log base format of this is Logb n = x. 

What are Logarithmic Functions?

• A logarithm is defined as the exponent or power to which a base must be raised to get some new number. It is a convenient approach to express large numbers.

• Through logarithm, the multiplication of large numbers we can easily resolve speedily. Some common properties of logarithm which proved multiplication and division of logarithms can even be written in the form of logarithm of addition or subtraction. 

Let’s take an example,

23=8, where 3 is the logarithm of 8 to base 2 or can be written as 3= log₂8.

Similarly, 10² = 100 which can be written as 2 = log₁₀100

Common or Briggsian logarithm is the logarithm with base 10.

Properties of Logarithm-

Given below are the four basic properties of logarithm which would help you to resolve problems based on logarithm.

Before deriving Log 1 value, let us discuss logarithm functions and their classifications. A logarithm function is an inverse function to an exponential function. In Mathematics Logarithm can be expressed in the following way:

If Log_ab = x, then a^x=b

where, “x” = log of a number 

“a” = base of a logarithm function.

Something you need to keep in mind! The variable “a” should always be a positive integer and not equal to 1.

Classification of Logarithm Function-

• There are two types of logarithmic functions.

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What do you Mean by Common Logarithmic Function?

• Common Logarithmic Function or Common logarithm is the logarithm with a base equal to 10.

• It is also known as the decimal logarithm because of its base.

• The common logarithm of x is denoted as log x.

• Example: log 100 = 2 (Since 10²= 100) 

What do you mean by Natural Logarithmic Function?

• The Natural Logarithmic Function is the logarithm with a base equal to the mathematical constant e.

• The value of e which is a mathematical constant is approximately equal to 2.7182818.

• The natural logarithm of x is written as log_ex.

• Example: log_e25= ln 25

What is the Value of Log 1? (A small explanation) How to find the value of log 1?

From the definition of logarithm function,  

We can write log_ab = x can be written in the form of an exponential form: a^x = b

Now to find the value of log 1, here the base is not defined.

 Let us consider the base of log 1 as 10. 

Therefore, we can write log 1 as log₁₀ 1.≠

From the logarithm definition, the value of a = 10 and the value of b = 1. 

Such that, log₁₀ x = 1

Now by the logarithm rule, the above expression can be rewritten:

10^x = 1

As we know, when any number is raised to the power 0 it is equal to 1.

Thus, 10 raised to the power 0 makes the above-written expression true. So, 10⁰ = 1

This is a condition for all the base values of log, where the base raised to the power 0 = 1

Therefore, we can conclude the value of log 1 is zero, where a can be any positive value (a≠1)

log₁₀1 = 0

How to Find the Value of Ln 1?

Similarly, the natural logarithm value of 1( Ln 1) can be represented by,

Ln(b) = log_e (b)

Therefore, Ln(1) = loge(1) or e^x = 1

Therefore, e⁰ = 1

Hence, we can conclude that Ln(1) = log_e(1) = 0

Log Value from 1 to 10

Table showing Ln Values from 1 to 10. Log Values From 1 to 10 to the Base e are Given Below -

Questions to be Solved on Log 1

Question 1)   Solve for y in log₂ y =6

Solution: The logarithm function of the above function can be written as 2⁶ = y

 Therefore, 2⁵ =2 x 2 x 2 x 2 x 2 x 2 =64 or y = 64

Question 2)   Calculate the value of x in 7^x =1000.

Solution: Taking common logarithms on both the sides, and applying the property of the logarithm of a power, 

7^x =1000 can be written as,

7^x=10³

Log 7^x = log 10³

x log7 = 3 log 10

x log7 = 3 .1 (Using the log 10 value which is 1)

x log7 = 3 

x log 7 log 7  =  3log 7 

Therefore, x = 3log 7

Fun Facts -

Using Logarithms in Daily Life

• Earthquakes that occur in high seismic zones are recorded on the seismograph. They are measured using a device called the “Richter Scale”. To read the recording on the Richter scale, one should use logarithms and know all the laws related to it. 

• Logarithms are used to determine the pH value of acids, bases and salts and also to compare them.

• Logarithms are used to measure complex values and represent them in the simple form. 

• Sound intensity from a source is measured by the concept of loudness. To measure the “loudness”, we use logarithmic values.