Exponential and Logarithmic Functions#
Definition of an Exponential Function
The function defined by
is called an exponential function with base \(b\) and exponent \(x\).
Properties of Exponential Functions
For a function \(y = b^x\), the following properties hold:
Its domain is \((-\infty,\infty)\).
Its range is \((0,\infty)\).
Its graph passes through the point \((0,1)\) (i.e., \(b^0 = 1\)).
It is continuous on \((-\infty,\infty)\).
It is increasing on \((-\infty,\infty)\) if \(b>1\) and decreasing on \((-\infty,\infty)\) if \(0<b<1\).
Definition of a Logarithmic Function
is called the logarithmic function with base \(b\) and is defined by
Properties of Logarithmic Functions
For a function \(y = \log_{b}(x)\), the following properties hold:
Its domain is \((0,\infty)\).
Its range is \((-\infty,\infty)\).
Its graph passes through the point \((1,0)\) (i.e., \(\log_b(1) = 0\)).
It is continuous on \((0,\infty)\).
It is increasing on \((0,\infty)\) if \(b>1\) and decreasing on \((-\infty,\infty)\) if \(0<b<1\).