# Introduction to Logarithmic Functions (Part 2)

(This page is Part 2. Click here for Part 1.)

## Properties that All Logarithmic Functions Share

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: $\,\text{dom}(f)=(0,\infty)$
- The range is the set of all real numbers: $\,\text{ran}(f)=\mathbb{R}\,$
- The graph crosses the $x$-axis at $\,x = 1$
- The graph passes both the vertical and horizontal line test

### 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, sending each point on the graph to its $x$-value, then all positive $x$-values will be hit.

Logarithms only know how to act on positive inputs.

For basic information on the domain and range of a function, you may want to review: Domain and Range of a Function

Having trouble understanding the expression ‘$\,(0,\infty)\,$’? Then, you may want to review Interval and List Notation.

### 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,
sending each point on the graph
to its $y$-value,
then *all*
$y$-values will be hit.

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,
which makes them particularly
valuable in modeling slowly-changing behavior.

### The Graph Crosses the $x$-axis at $\,x=1$

For allowable values of $\,b\,$:

$$ \cssId{s34}{\log_b\,1 \overset{\text{always}}{\ \ \ =\ \ \ } 0\ ,} \ \ \ \cssId{s35}{\text{ since }\ \ \ b^0 \overset{\text{always}}{\ \ \ =\ \ \ } 1} $$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.

### The Graph Passes Both the Vertical and Horizontal Line Test

Vertical Line Test:
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 $x$-value

that has more than one $y$-value

Horizontal Line Test:
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 *one-to-one correspondence*
between the inputs and outputs.
Functions with this property are
called *one-to-one functions*.