Value of Log 1

Log 1 Value

• In Mathematics, most of the researchers use 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 these both subjects contend with large numbers.

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

What are Logarithm 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,

$2^{3}$ = 8, where 3 is the logarithm of 8 to base 2 or can be written as 3= log28.

In a similar way, 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.

 Logb(mn) = Logb m + Logb nThis property of logarithm says that the multiplication of two logarithm values is equivalent to the addition of the individual logarithm. Logb (m/n) = Logb m - Logb nThis property of logarithm says that the division of two logarithm values is equivalent to the subtraction of the individual logarithm. Logb (mn) = n logbmThe above property is known as the exponential rule of the logarithm. The logarithm of m along with the rational exponent is equivalent to the exponent times its logarithm. Logb m = loga m / loga When two numbers are divided with the same base, then the exponents will be subtracted.

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:

 REPRESENTATION OF A LOGARITHM  FUNCTION logab =x, then ax=b                            .

If Logab = x, then ax=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 equals 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 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 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 logex.

• Example: loge25= ln 25

How to find the value of log 1?

From the definition of logarithm function,

We can write logab = x can be written in the form of an exponential form: ax = 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:

10x = 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 value 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) = loge (b)

Therefore, Ln(1) = loge(1) or ex = 1

Therefore, e0 = 1

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

Log Value from 1 to 10

 Value of log Log 1 0 Log 2 0.3010 Log 3 0.4771 Log 4 0.6020 Log 5 0.6989 Log 6 0.7781 Log 7 0.8450 Log 8 0.9030 Log 9 0.9542 Log 10 1

Log Values From 1 to 10 to the Base e are Given Below -

 In (1) 0 In (2) 0.693147 In (3) 1.09861 In (4) 1.38629 In (5) 1.60944 In (6) 1.79176 In (7) 1.94591 In (8) 2.07944 In (9) 2.19722 In (10) 2.30259

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 7x=1000.

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

7x=1000 can be written as,

7x=103

Log 7x= 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 -

 The first man to use Logarithm in modern times was the German Mathematician, Michael Stifel (around 1487 -1567).The logarithm with base 10 is known as common or Briggsian, logarithms and can be written as log n. They are usually written as without base.