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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} is a positive integer, this just follows from repeated application of the above-mentioned additive property of the logarithm. For Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p = \frac{1}{n}} , note that the statement follows by observing that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ln(x) = \ln((x^{1/n})^n) = n\ln(x^{1/n})} . Thus, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p\ln(x) = \ln(x^p)} for all rational numbers Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} . The statement must therefore hold for all reals Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} by continuity.

• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ln(1) = 0}

Proof: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ln(1) = \ln(1\cdot 1) = \ln(1)+\ln(1)} , whence we must have Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ln(1)=0} .