Summary of Exponential and Logarithmic Functions

The exponential function $y=b^x$ is increasing if $b>1$ and decreasing if [latex]0
The logarithmic function $y=\log_b(x)$ is the inverse of $y=b^x$ . Its domain is $(0,\infty)$ and its range is $(−\infty,\infty)$ .
The natural exponential function is $y=e^x$ and the natural logarithmic function is $y=\ln x=\log_e x$ .
Given an exponential function or logarithmic function in base $a$ , we can make a change of base to convert this function to any base $b>0, \, b \ne 1$ . We typically convert to base $e$ .
The hyperbolic functions involve combinations of the exponential functions $e^x$ and $e^{−x}$ . As a result, the inverse hyperbolic functions involve the natural logarithm.

Glossary

base: the number $b$ in the exponential function $f(x)=b^x$ and the logarithmic function $f(x)=\log_b x$

hyperbolic functions: the functions denoted $\sinh, \, \cosh, \, \tanh, \, \text{csch}, \, \text{sech}$ , and $\coth$ , which involve certain combinations of $e^x$ and $e^{−x}$

inverse hyperbolic functions: the inverses of the hyperbolic functions where $\cosh$ and $\text{sech}$ are restricted to the domain $[0,\infty)$ ; each of these functions can be expressed in terms of a composition of the natural logarithm function and an algebraic function

number e: as $m$ gets larger, the quantity $(1+(1/m))^m$ gets closer to some real number; we define that real number to be $e$ ; the value of $e$ is approximately 2.718282