Cambridge O Level Additional Mathematics · Syllabus 4037 · Logarithmic and Exponential Functions
Natural Logarithm
What is Natural Logarithm?
The logarithm to base e, written ln x, where e is the irrational constant 2.718 to three decimal places. It is the inverse function of the exponential function e to the power x, so ln of e to the power x equals x for every real x, and e to the power ln x equals x for every positive x. Its domain is x greater than zero, its range is all real numbers, it crosses the x-axis at the point one comma zero, it has the vertical asymptote x equals zero, and it is increasing throughout its domain.
This definition is part of the Logarithmic and Exponential Functions chapter in Cambridge O Level Additional Mathematics.
Natural Logarithm in context
The exponential function \(e^{x}\) and the natural logarithm \(\ln x\) are inverse functions, so \(\ln(e^{x})=x\) for every real \(x\) and \(e^{\ln x}=x\) for every positive \(x\). Their graphs are reflections of each other in the line \(y=x\), and their domains and ranges exchange: \(e^{x}\) accepts every real number and produces only positive values, with the horizontal asymptote \(y=0\), while \(\ln x\) accepts only positive numbers and produces every real value, with the vertical asymptote \(x=0\). That inverse relationship is why \(\ln\) is the tool that removes \(e\), and \(e\) is the tool that removes \(\ln\).