Difference Between Log and Ln

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.

Log is defined for base 10 whereas, ln is defined for the base e. Example- log of base 2 is written as log₂ while log of base e is represented as log_e= ln (natural log).

The logarithm which is defined as the power to which the base is e that has to be raised to obtain a number is called its log number of the natural logarithm. ‘e’ is the exponential function.

Definition of Log

The logarithm in mathematics is the inverse function of exponentiation. In other words, a log is defined as the power to which a number must be raised such that we get the other number. This is also known as the logarithm of base 10 or the common logarithm. The general form of the logarithm is:

log_a (y) = x

It is also written as

a^x = y

Properties of Logarithm

• Log_b (mn)= log_b m + log_b n
• Log_b (m/n)= log_b m – log_bn
• Log_b (mn) = n log_b m
• Log_b m = log_a m/log_a b

Definition of ln

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.

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 differences between Log and natural log are given below in a tabular form:

	log	ln
1.	Log generally refers to a logarithm to the base 10	Ln generally refers to a logarithm to the base e
2.	Also known as the common logarithm	Also called the natural logarithm
3.	The common log is represented as log10 (x)	The natural log is represented as loge (x)
4.	The exponential form for this log is 10x = y	It has the exponential form as ex=y
5.	The interrogative statement for the common logarithm is “At which number should we raise 10 to get y?”	The interrogative statement for the natural logarithm is “At which number should we raise Euler’s constant number to get y?”
6.	It is mostly used in physics as compared to ln	It has much less use in physics
7.	It is Represented as log base 10 in maths	This is represented as log base e.

Sample Questions

Question 1. Solve for a in log₂ a = 5

Solution:

The logarithm function of the above function can be written as  2⁵=a

Therefore,  2⁵= 2 x 2 x 2 x 2 x 2 =32  or y = 32

Question 2. Simplify log(75).

Solution:

We will use the Log and ln rules we have discussed. Since we know that the number 75 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(75) = 1.87506126339 or log(75) = 1.87 rounded to two decimal places. 

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