Exponential and Logarithmic Functions#

Definition of an Exponential Function

The function defined by

\[f(x) = b^x \qquad (b > 0, b \neq 1)\]

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

\[\begin{align*} f(x) = \log_b(x) & \qquad (b>0,b \neq 1) \end{align*}\]

is called the logarithmic function with base \(b\) and is defined by

\[y = \log_b(x) ~~~~ \hbox{ if and only if } ~~~~ x = b^y.\]

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 \(0<b<1\).