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Difference Between Log and Ln for JEE Main 2024

JEE Main

Difference Between

Maths

Difference Between Log And Ln

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Log Definition

Log and Ln stand for Logarithm and Natural Log respectively. Logarithms are essential for solving equations where an unknown variable appears as the exponent of some other quantity. They are significant in many branches of mathematics and scientific subjects and are used to solve problems involving compound interest, which is broadly related to finance and economics.

In Mathematics, the logarithm can be defined as the inverse function of exponentiation.
In simpler words, the logarithm can be defined as a power to which a number must be raised in order to get any other number.
It is also known as the logarithm of base 10, or common logarithm.
The general form of a logarithm can be denoted as:

logₐ (y) = x

The above - given form can also be written as:

\[a^{x} = y\]

In this article we are going to discuss what is log, what is ln in math, Log and ln rules , the difference between Log and Ln x , difference between log and natural log and difference between log and ln graph.

Given below are the four basic properties of logarithm which will help you to easily solve problems based on logarithm.

Properties of Logarithm

Logb(mn) = Logb m + Logb n

This property of logarithm denotes that the multiplication of two logarithm values is equivalent to the addition of the individual logarithm.

Logb (m/n) = Logb m - Logb n

This property of logarithm says that the division of two logarithm values is equivalent to the subtraction of the individual logarithm.

Logb (mn) = n logbm

The 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.

Log Value from 1 to 10

Log	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

What is Ln in Maths?

Ln is called the natural logarithm. It is also called the logarithm of the base e. Here, the constant e denotes a number that is a transcendental number and an irrational which is approximately equal to the value 2.71828182845. The natural logarithm (ln) can be represented as ln x or \[\log_{e}x\].

Let’s go Through the Different Rules of Ln

Quotient Rule

ln(x/y) is equal to ln(x) - ln(y)

The natural log of the division of x and y is equal to the difference of the ln of x and ln of y.

Example: ln(10/5) = ln(10) - ln(5)

Reciprocal Rule

ln(1/x) is equal to − ln(x)

The natural log of the reciprocal of x is equal to the opposite of the ln of x.

Example: ln(⅓) equals -ln(3)

Power Rule

\[\ln (x^{y})\] is equal to \[y \times \ln x\]

The natural log of x raised to the power of y is equal to y times the ln of x.

Example: ln(4²) equals to 2 * ln(4)

We have discussed the log and ln rules above.

Log values from 1 to 10 to the base e are given below-

Table Showing Ln Values From 1 to 10.

In (1)	0
In (2)	0.693147
In (3)	1.098612
In (4)	1.386294
In (5)	1.609438
In (6)	1.791759
In (7)	1.94591
In (8)	2.079442
In (9)	2.197225
In (10)	2.302585

Difference Between Log and ln Graph

These graphs will show you the difference between log and ln graph.

(Images will be uploaded soon)

Let’s discuss some of the key differences Between Log and Ln:

To solve logarithmic problems,one must know the difference between log and natural log. Having a key understanding of the exponential functions can also prove helpful in understanding different concepts. Some of the important difference between Log and natural log are given below in a tabular form:

Difference Between Log and Ln x

Log	Ln
Log generally refers to a logarithm to the base 10.	Ln basically refers to a logarithm to the base e.
This is also known as a common logarithm.	This is also known as a natural logarithm.
The common log can be represented as log10 (x).	The natural log can be represented as loge (x).
The exponent form of the common logarithm is written as \[10^{x} = y\]	The exponent form of the natural logarithm can be written as \[e^{x} = y\]
The interrogative statement for the common logarithm is written as “At which number should we raise 10 to get y?”	The interrogative statement for the natural logarithm is written as“At which number should we raise Euler’s constant number to get y?”
The log function is more widely used in physics when compared to ln.	As logarithms are usually taken to the base in physics, ln is used much less.
Mathematically, it can be represented as log base 10.	Mathematically, ln can be represented as log base e.

Questions to be Solved:

Question 1) Solve for y in log₂ y =6

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

Therefore, \[2^{6}\] = 2 x 2 x 2 x 2 x 2 x 2 =64 or y = 64

Question 2) Simplify log(98).

Answer) We will use the Log and ln rules we have discussed. Since, we know that the number 98 is not a power of 10 (the way that 100 was), So we can find the value by plugging this into a calculator, remembering to use the "LOG" key (not the "LN" key), and we get log(98) = 1.99122607569..., or log(98) = 1.99, rounded to two decimal places.

Fun Facts

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

FAQs on Difference Between Log and Ln for JEE Main 2024

1. Is Log Base 10 the Same as Ln?

A logarithm is a number that is written as \[\log_{b}(x)\], and it is equal to the number that we need to raise b to in order to get x. In mathematics, some logarithms show up more often than others, and we classify these logarithms as special types of logarithms by the value of their base.

2. Does Ln Mean Log?

ln(x) means the base e logarithm; it can also be written as \[\log _{e}(x).\ln (x)\].

3. How should Log be used?

When logarithms are added, they should be multiplied. Let’s learn about it through an example:

\[\log_{2}{x}+ \log_{2}{y}= \log_{2}{xy}\]

Another example is log2128. We may not know what 2 to whose power is 128 from the top of the head so basically, we can split this into two smaller logarithms using this property. Here is how it’s split up.

\[\log_{2}{128} = \log_{2}{32} + \log_{2}{4}\]

After finding the logs through the exponents, this is our final answer.

\[\log_{2}{128} = \log_{2}{32} + \log_{2}{4}\]

\[\log_{2}{128} = 5+2\]

\[\log_{2}{128} = 7\]

This is the quickest and smartest way of solving a logarithm. You should keep in mind that this method can be used only when the bases of both the logarithms are similar.