In this section we continue to explore logarithms, but now with an emphasis on the family of logarithmic functions.
Some academic disciplines use the natural log a lot, but the common log not so much.
These disciplines often use $\,\log\,$ to mean log base e, instead of log base ten.
Be sure to check what notation is being used whenever you start communicating with someone new.
Furthermore, some mathematicians (this author included) tend to call more general functions
where a logarithm is acting on variable input a logarithmic function;
say,
$\,y=\ln (2x3)\,$ or
$\,y=7\log(5x)\,$.
For the purposes of this section, however,
the phrase logarithmic function refers
only to functions of the form $\,y=\log_b\,x\,$.
There are two basic shapes to the graphs of logarithmic functions,
depending on whether the base is greater than $\,1\,$,
or between $\,0\,$ and $\,1\,$:
$\,y=\log_b\,x\,$ for $\,b\gt 1\,$  $\,y=\log_b\,x\,$ for $\,0\lt b\lt 1\,$ 
increasing functions  decreasing functions 
PROPERTIES OF THE GRAPH for $\,b\gt 1\,$
An increasing function has the following property:
you may want to review Basic Models You Must Know. 
$y = \log_b\,x\,$ for $\,b\gt 1$
increasing functions as $\,x\rightarrow\infty\,$, $\,y\rightarrow\infty\,$ as $\,x\rightarrow 0^+\,$, $\,y\rightarrow \infty\,$ 
PROPERTIES OF THE GRAPH for $\,0 \lt b\lt 1\,$
A decreasing function has the following property:

$y = \log_b\,x\,$ for $\,0 \lt b\lt 1$
decreasing functions as $\,x\rightarrow\infty\,$, $\,y\rightarrow \infty\,$ as $\,x\rightarrow 0^+\,$, $\,y\rightarrow \infty\,$ 
Let $\,f(x)=\log_b\,x\,$,
where $\,b\,$ is a positive number not equal to $\,1\,$,
and $\,x\gt 0\,$.
For all (allowable) bases $\,b\,$, logarithmic functions share the following properties:
THE DOMAIN IS THE SET OF POSITIVE NUMBERS:
$\text{dom}(f) = (0,\infty)$
If the graph of a logarithmic function is ‘collapsed’ into the $\,x$axis, Logarithms only know how to act on positive inputs.
For basic information on the domain and range of a function,
Having trouble understanding the expression ‘$\,(0,\infty)\,$’? 

THE RANGE IS THE SET OF ALL REAL NUMBERS:
$\text{ran}(f)=\mathbb{R}$
If the graph of a logarithmic function is ‘collapsed’ into the $\,y$axis, In particular, even though increasing logarithm curves rise very slowly for large inputs, they WILL eventually reach any desired output, no matter how big (and positive) it may be! And, even though decreasing logarithm curves fall very slowly for large inputs, they WILL eventually reach any desired output, no matter how big (and negative) it may be!
Indeed, the fact that logarithmic functions increase/decrease very slowly for large inputs is an important
feature of their graphs,


THE GRAPH CROSSES THE $\,x\,$AXIS AT $\,x=1\,$
For allowable values of $\,b\,$: $$ \cssId{sb34}{\log_b\,1 \overset{\text{always}}{\ \ \ =\ \ \ } 0\ ,} \ \ \ \cssId{sb35}{\text{ since }\ \ \ b^0 \overset{\text{always}}{\ \ \ =\ \ \ } 1} $$ So, when the input is $\,1\,$ to the function $\,\log_b\,$, the output is $\,0\,$. Thus, the point $\,(1,0)\,$ lies on the graph of every logarithmic function. 

Imagine a vertical line sweeping through a graph, checking each allowable $\,x$value:
if it never hits the graph at more than one point, then the graph is said to pass the vertical line test.
All functions pass the vertical line test, since the function property is that
each input has exactly one output.
passes the vertical line test: each $\,x$value has only one $\,y$value all functions pass the vertical line test 
fails the vertical line test: there exists an xvalue that has more than one yvalue 
Imagine a horizontal line sweeping through a graph, checking each allowable $\,y$value:
if it never hits the graph at more than one point, then the graph is said to pass the horizontal line test.
Some functions pass the horizontal line test, and some do not.
passes the horizontal line test: each $\,y$value has only one $\,x$value all logarithmic functions pass the horizontal line test 
fails the horizontal line test: there exists a $\,y$value that has more than one $\,x$value some functions fail the horizontal line test 
Thus, logarithmic functions have a wonderful property:
each input has exactly one output (passes the vertical line test),
and
each output has exactly one input (passes the horizontal line test).
For such functions, you can think of the inputs/outputs as being connected with strings:
pick up any input, and follow its ‘string’ to the unique corresponding output;
pick up any output, and follow its ’string’ to the unique corresponding input.
That is, there is a onetoone correspondence between the inputs and outputs.
Functions with this property are called onetoone functions.
On this exercise, you will not key in your answer. However, you can check to see if your answer is correct. 
PROBLEM TYPES:
