Difference Between Log and Ln

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 log₂ while log of base e is represented as log_e= ln (natural log).

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_e x.

If e^x = z  ⇔ log_e (z) = x

Properties of Logarithm

Some properties of logarithm are:

• 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

Differences Between Log and Ln

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

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

ln(x) = \frac{\log_{10}(x)}{\log_{10}(e)}

To convert ln_e(x) into Log:

\log_{10}(x) = \frac{\ln_{e}(x)}{\ln_{e}(10)}

Also read,

• Log Formulas
• Application of Logarithm