Question

How does taking natural log ($\ln$) change units ?

Answer 1

We know that the natural logarithm $\ln$ of any number is it’s logarithm to the base of the mathematical constant $e$, where $e$ is an irrational number and a number that is not the root of any integer, which is approximately equal to $2.718281828459$. The natural logarithm of $x$ is generally written as $\ln x$,${{\log }_{e}}x$, or sometimes, if base is implicit (i.e. relation of the form $R({{x}_{1}},....,{{x}_{n}})=0$ where $r$ is the function of several variables), simply $\log x$.

Complete answer:
In most cases natural logarithm is taken of dimensionless quantities. Like in physics the argument of the natural logarithm is always a ratio of two quantities having the same dimensions. In mathematics natural logarithm is taken simply as a number. No dimensions are involved.

But sometimes people write expressions involving natural logs of numbers with dimensions. Those can’t be evaluated. Taking the natural logarithm of any quantity doesn’t really change the dimension of the quantity but it does change the numerical value so the actual quantity and the natural logarithm of that quantity should be considered different units.

Note: There is a difference between logarithm and natural logarithm. In logarithm the logarithm with base $10$ and is also called common logarithm, whereas in natural logarithm the logarithm is of base ‘$e$’. Where $e$ is the exponential function. The common logarithm is represented as ${{\log }_{10}}x$ whereas the natural logarithm is represented as ${{\log }_{e}}x$. The exponent form of the common logarithm is ${{10}^{x}}=y$ whereas the exponent form of the natural logarithm is ${{e}^{x}}=y$. Mathematically common logarithm is represented as log base $10$ and mathematically natural logarithm is represented as log base $e$.