Natural logarithm

red: natural logarithm

The natural logarithm, $\ln(x)$ is the inverse of the function $e^{x}$. In other words, if $y=e^{x}$, we define $\ln(y)=x$.

The natural logarithm has some interesting properties that follow from the multiplicative properties of $e^{x}$. The natural logarithm is also particularly useful in calculating interest.

Properties of the Logarithm

• $\ln(ab)=\ln(a)+\ln(b)$ for all positive reals $a,b$.

Proof: If $a,b>0$, we can write $a=e^{x}$ and $b=e^{y}$. It follows that $\ln(ab)=\ln(e^{x}e^{y})=\ln(e^{x+y})$. By definition, $\ln(e^{x+y})=x+y=\ln(e^{x})+\ln(e^{y})$. This last expression, of course, is $\ln(a)+\ln(b)$.

• $\ln(x^{p})=p\ln(x)$

Proof: If $p$ is a positive integer, this just follows from repeated application of the above-mentioned additive property of the logarithm. For $p={\frac {1}{n}}$, note that the statement follows by observing that $\ln(x)=\ln((x^{1/n})^{n})=n\ln(x^{1/n})$. Thus, $p\ln(x)=\ln(x^{p})$ for all rational numbers $p$. The statement must therefore hold for all reals $p$ by continuity.

• $\ln(1)=0$

Proof: $\ln(1)=\ln(1\cdot 1)=\ln(1)+\ln(1)$, whence we must have $\ln(1)=0$.