Key Terms

asymptote

A line which a graph of a function approaches closely but never touches.

common logarithmic function

The function $f\left(x\right)=\text{log}x$ is the common logarithmic function with base$10,$ where $x>0.$

$$y=\text{log}x\text{is equivalent to}x={10}^y$$

exponential function

An exponential function, where $a>0$ and $a\ne 1,$ is a function of the form $f\left(x\right)=a^x.$

logarithmic function

The function $f\left(x\right)={\text{log}}_{a}x$ is the logarithmic function with base $a,$ where $a>0,$$x>0,$ and $a\ne 1.$

$$y={\text{log}}_{a}x\text{is equivalent to}x=a^y$$

natural base

The number e is defined as the value of ${\left(1+\frac{1}{n}\right)}^n,$ as n gets larger and larger. We say, as n increases without bound, $e\approx \mathrm{2.718281828..}.$

natural exponential function

The natural exponential function is an exponential function whose base is e: $f\left(x\right)=e^x.$ The domain is $\left(\text{−}\infty ,\infty \right)$ and the range is $\left(0,\infty \right).$

natural logarithmic function

The function $f\left(x\right)=\text{ln}x$ is the natural logarithmic function with base $e,$ where $x>0.$

$$y=\text{ln}x\text{is equivalent to}x=e^y$$

one-to-one function

A function is one-to-one if each value in the range has exactly one element in the domain. For each ordered pair in the function, each y-value is matched with only one x-value.