Summary of Exponential and Logarithmic Functions

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

Glossary

base: the number [latex]b[/latex] in the exponential function [latex]f(x)=b^x[/latex] and the logarithmic function [latex]f(x)=\log_b x[/latex]

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

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

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