Section IN Exp and Log Functions

Exponential and Logarithmic Functions

Introduction: Of all the functions that are studied, after algebraic functions. the simplest class of core functions is the class of exponential functions. These functions are characterized by the appearance of a variable quantity in the exponent with a constant base. Paired with these functions are their inverses, the logarithmic functions.

For  example, the function defined by assigning the value $f(x) = exp_2(x) =2^x$ to the number $x$, also represented symbolically as $f: x \rightarrow exp_2(x) = 2^x$, or verbally  by saying "to find $f(x) = exp_2(x)$, raise the number $2$ to the $x$ power", is described as an exponential function. The number $2$ is described as the base of this exponential function. This  function is visualized  by the  mapping diagram, ELF1. 

The inverse function for $f(x) = 2^x = exp_2(b)$ is called the logarithm function, traditionally denoted $log_2$. This function is also represented symbolically as $f: x \rightarrow log_2(x)$, or verbally  by saying "to find $log_2(x)$, find the number $y$ where $2$ to the $y$ power is $x$". The domain of $log_2$ is $(0,\infty)$ and it satisfies the inverse function defining property:

For all $a \in R$ and $b \in (0,\infty )$,
$log_2(b)=a$ if and only if $exp_2(a) = 2^a = b$,
or,
$log_2(exp_2(a))=log_2(2^a)=a$ and  $exp_2(log_2(b)) = 2^{log_2(b)}=b$.

The number $2$ is described as the base of this logarithmic function. This  function is visualized  by the  mapping diagram, ELF2.

Mapping Diagram for f(x) =
                      2^x Mapping
                      Diagram for f(x) = log_2(x)    
Mapping Diagram ELF1
Mapping Diagram ELF2

This section will provide examples, explanations, exercises and problems that will help students use the power of the mapping diagram  along with the three other tools (equations, tables , and graphs) to understand exponential and logarithmic functions.
    6    Exponential and Logarithmic Functions (1 week)
    6.1                 Linear vs geometric growth
    6.1.1                 linear composition
    6.1.2            Inverses
    6.1.3    Logarithms
    6.1.3.1 Natural Exponential and Logarithmic Functions
    6.1.4         Domain
    6.1.5         Linear Composition
    6.1.6         Asymptotic Behavior
    6.1.7         Algebra Properties of Logs and Exponential Functions

Exponential Function Definition


Logarithmic Function Definition


ELF.ELFIExponential and Logarithmic Functions are Important. (Not Yet Done)
Example ELF.0 The First Exponential and Logarithmic Function Example.
This example presents the exponential function $f(x) = 2^x$ and the logarithmic function $g(x) = log_2(x)$ with a table of data,  a graph and a mapping diagram for each. The "base" in both examples is $2$.

Treatment of exponential and logarithmic functions and their graphical interpretation are familiar. [See wikipedia.org/wiki : exponential_function  /  logarithmic function]
They appear in every textbook that deals at all with intermediate algebra - from algebra II to beginning calculus. What is missing is a balanced treatment using mapping diagrams to reinforce the function aspect of visualization.  That will be emphasis of this section. 

Comparisons will be made when appropriate to graphs- but we will develop the basic concepts for exponential and logarithmic functions with mapping diagrams. The end of this section includes some powerful and different ways to think about exponential and logarithmic functions and the ways they are represented algebraically.

Subsection ELF.CELF Core Exponential and Logarithmic Functions 

Subsection ELF.NEL  Natural Exponential and Logarithmic Functions

Subsection ELF.ID Increasing/Decreasing for Exponential and Logarithmic Functions

Subsection ELF.AP  Algebraic Properties of Exp and Log Functions
ELF.AP.Proofs  Proofs of Algebraic Properties of Exp and Log Functions

Subsection ELF.LCOMP  Linear Composition with Core Exponential and Logarithmic Functions

Subsection ELF.COMP Symmetry and Composition of Exponential and Logarithmic Functions

Subsection ELF.INV Inverses for Exponential and Logarithmic Functions

Subsection ELF.SEq Solving Exponential and Logarithmic Equations

Subsection ELF.APP Exponential and Logarithmic Function Applications (Not Yet Done)


Exercises