Exponential and Logarithmic Functions

Exponential Functions

Any function of the form $f(x)=b^x,whereb>0,b\ne 1$ is an exponential function with base $b$  and exponent $x$ . Exponential functions have constant bases and variable exponents. Note that a function of the form $f(x)=x^b$ for some constant $b$ is not an exponential function but a power function.

 

※ Evaluating Exponential Functions

If $x$ is a positive integer, then we define $b^x=b\cdot b\cdots b$ (with $x$  factors of $b$ ). If $x$ is a negative integer, then $x=-y$ for some positive integer $y$, and we define $b^x=b^{-y}=1/b^y$. Also $b^0$ is defined to be $1$.

If $x$ is a rational number, then $x=p/q$ where  $p$ and  $q$ are integers and $b^x=b^{p/q}=\sqrt[q]{b^p}$. 

However, how is $b^x$ defiend if $x$ is an irrational number? This is too complex a question for us to answer fully right now; however, we can make an approximation.

 

※ Graphing Exponential Functions

For any base $b>0,b\ne 1$, the exponential function $f(x)=b^x$ is defined for all real numbers $x$ and $b^x>0$. Therefore, the domain of $f(x)=b^x$ is $(-\mathrm{\infty },\mathrm{\infty })$ and the range is $(0,\mathrm{\infty })$.

 

The Number $e$

Examine the behavior of $(1+1/m{)}^m$ as $m\to \mathrm{\infty }$

It appears that $(1+1/m{)}^m$ is approaching a number between 2.7 and 2.8. We call thisnumber e. To six decimal places of accuracy, $$e\approx 2.718282$$

we call the function $f(x)=e^x$ the natural exponential function.

 

 

 

Logarithmic Functions

The logarithmic functions are inverse functions of exponential functions. For any base $b>0,b\ne 1$, the logarithmic function with base $b$, denoted, $lo{g}_b$ has domain $(0,\mathrm{\infty })$ and range $(-\mathrm{\infty },\mathrm{\infty })$, and satisfies $$lo{g}_b(x)=yifandonlyif{b}^y=x$$

 

 

 

Hyperbolic Functions

The hyperbolic functions are defined in terms of certain combinations of $e^x$ and $e^{-x}$.

The name cosh rhymes with “gosh,” whereas the name sinh is pronounced “cinch.” Tanh, sech, csch, and coth are pronounced “tanch,” “seech,” “coseech,” and “cotanch,” respectively.

But why are these functions called hyperbolic functions? To answer this question, consider the quantity $cos{h}^{2}t-sin{h}^{2}t$. Using the definition of cosh and sinh, we see that

This identity is the analog of the trigonometric identity $co{s}^{2}t+si{n}^{2}t=1$. Here, given a value $t$, the point $(x,y)=(cosht,sinht)$ lies on the unit hyperbola $x^2-y^2=1$.

 

※ Graphs of Hyperbolic Functions

 

 

※ Identities Involving Hyperbolic Functions

 

※ Inverse Hyperbolic Functions

From the graphs of the hyperbolic functions, we see that all of them are one-to-one except coshx and sechx. If we restrict the domains of these two functions to the interval [0, ∞), then all the hyperbolic functions are one-to-one, and we can define the inverse hyperbolic functions. Since the hyperbolic functions themselves involve exponential functions, the inverse hyperbolic functions involve logarithmic functions.