Difference Between Log and Ln

Q: What is the difference between log and natural log?

The difference between log and natural log is their base: log uses base 10, and natural log (ln) uses the number e ≈ 2.718 as its base.

Q: Is log x the same as ln x?

No, "log(x)" is not same as "ln(x)" as the natural log(ln) uses the mathematical constant e as its base whereas, the common log(log) uses 10 as its base.

Last Updated : 18 Dec, 2024

Logarithms(log) and natural logarithms(ln) are fundamental mathematical concepts that simplify complex calculations involving exponential relationships Logarithms are essential for solving equations where an unknown variable appears as the exponent of some other quantity.

A logarithm can have any positive base, with the common log having a base 10, while the natural log exclusively uses the mathematical constant (e≈2.718) as its base.

Example– log of base 2 is written as log₂while log of base e is represented as log_e= ln (natural log).

Log vs ln

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:

If 10^x = y ⇔ log₁₀(y) = x

Definition of Natural log (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_ex.

If e^x = z ⇔ log_e(z) = x

Properties of Logarithm

Some properties of logarithm are:

Log_b (mn)= log_b m + log_bn
Log_b (m/n)= log_b m – log_bn
Log_b (mn) = n log_bm
Log_b m = log_a m/log_ab

Differences Between Log and Ln

Some of the important differences between Log and natural log are given below:

log	ln
Log generally refers to a logarithm to the base 10	Ln generally refers to a logarithm to the base e
Also known as the common logarithm	Also called the natural logarithm
The common log is represented as log₁₀(x)	The natural log is represented as log_e(x)
The exponential form for this log is 10^x = y	It has the exponential form as e^x= y
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?”
It is mostly used in physics as compared to ln	It has much less use in physics
It is represented as log base 10 in math	This is represented as log base e.

Step to Convert Log into ln

To convert the common log ( log) into natural log(ln) we can simply use the base change formula to change the base from 10 to e.

Base change formula for Logarithm: log_ba = log_ca / log_cb

To convert Log₁₀(x) into ln

[Tex]ln(x) = \frac{\log_{10}(x)}{\log_{10}(e)}[/Tex]

To convert ln_e(x) into Log:

[Tex]\log_{10}(x) = \frac{\ln_{e}(x)}{\ln_{e}(10)}[/Tex]

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