Difference Between Log and Ln

Log Definition

  • 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 / log


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 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 L

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 * 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 10x =y

The exponent form of the natural logarithm can be written as ex =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, 25 =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 nice neat power of 10 (the way that 100 was),so we cannot be clever with exponents to arrive at an exact answer. 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.

FAQ (Frequently Asked Questions)

Question 1) Is Log Base 10 the Same as Ln?

Answer)Special Logarithms:

A logarithm is a number that is written as logb(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.

Question 2) What is ln in math?

Answer)Let’s know what is Ln equal to,the natural logarithm of any number can be defined as its logarithm to the base of the mathematical constant that is e, where e is equal to an irrational and transcendental number which is approximately equal to the value 2.718281828459. The natural logarithm of x can generally be written as ln x, logex, or sometimes, if the base e is implicit, simply log x.

Question 3) Why do we use in Instead of Log?

Answer)So the natural logarithm is the logarithm in base , which means that all logarithms behave the same, regardless of the base. There is just a scale factor. So, when you don't need to specify a base, then you can just say without subindex.

Question 4)Does Ln Mean Log?

Answer)Usually log(x) refers to the base 10 logarithm; it can also be written as log10(x) .Whereas ln(x) means the base e logarithm; it can also be written as loge(x) . ln(x).