Cambridge O Level Additional Mathematics · Syllabus 4037 · Logarithmic and Exponential Functions
Exponential Function
What is Exponential Function?
A function of the form f of x equals a to the power x, where the base a is a fixed positive constant not equal to one and the exponent is the variable. Its domain is every real number and its range is strictly positive values only, so its graph lies entirely above the x-axis and has the horizontal asymptote y equals zero. Every exponential function passes through the point zero comma one, because any non-zero base raised to the power zero equals one. When the base is greater than one the function increases and models growth; when the base lies strictly between zero and one the function decreases and models decay.
This definition is part of the Logarithmic and Exponential Functions chapter in Cambridge O Level Additional Mathematics.
Exponential Function 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 \(\ln\).