# 6.4 Graphs of logarithmic functions  (Page 2/8)

 Page 2 / 8

Given a logarithmic function, identify the domain.

1. Set up an inequality showing the argument greater than zero.
2. Solve for $\text{\hspace{0.17em}}x.$
3. Write the domain in interval notation.

## Identifying the domain of a logarithmic shift

What is the domain of $\text{\hspace{0.17em}}f\left(x\right)={\mathrm{log}}_{2}\left(x+3\right)?$

The logarithmic function is defined only when the input is positive, so this function is defined when $\text{\hspace{0.17em}}x+3>0.\text{\hspace{0.17em}}$ Solving this inequality,

The domain of $\text{\hspace{0.17em}}f\left(x\right)={\mathrm{log}}_{2}\left(x+3\right)\text{\hspace{0.17em}}$ is $\text{\hspace{0.17em}}\left(-3,\infty \right).$

What is the domain of $\text{\hspace{0.17em}}f\left(x\right)={\mathrm{log}}_{5}\left(x-2\right)+1?$

$\left(2,\infty \right)$

## Identifying the domain of a logarithmic shift and reflection

What is the domain of $\text{\hspace{0.17em}}f\left(x\right)=\mathrm{log}\left(5-2x\right)?$

The logarithmic function is defined only when the input is positive, so this function is defined when $\text{\hspace{0.17em}}5–2x>0.\text{\hspace{0.17em}}$ Solving this inequality,

The domain of $\text{\hspace{0.17em}}f\left(x\right)=\mathrm{log}\left(5-2x\right)\text{\hspace{0.17em}}$ is $\text{\hspace{0.17em}}\left(–\infty ,\frac{5}{2}\right).$

What is the domain of $\text{\hspace{0.17em}}f\left(x\right)=\mathrm{log}\left(x-5\right)+2?$

$\left(5,\infty \right)$

## Graphing logarithmic functions

Now that we have a feel for the set of values for which a logarithmic function is defined, we move on to graphing logarithmic functions. The family of logarithmic functions includes the parent function $\text{\hspace{0.17em}}y={\mathrm{log}}_{b}\left(x\right)\text{\hspace{0.17em}}$ along with all its transformations: shifts, stretches, compressions, and reflections.

We begin with the parent function $\text{\hspace{0.17em}}y={\mathrm{log}}_{b}\left(x\right).\text{\hspace{0.17em}}$ Because every logarithmic function of this form is the inverse of an exponential function with the form $\text{\hspace{0.17em}}y={b}^{x},$ their graphs will be reflections of each other across the line $\text{\hspace{0.17em}}y=x.\text{\hspace{0.17em}}$ To illustrate this, we can observe the relationship between the input and output values of $\text{\hspace{0.17em}}y={2}^{x}\text{\hspace{0.17em}}$ and its equivalent $\text{\hspace{0.17em}}x={\mathrm{log}}_{2}\left(y\right)\text{\hspace{0.17em}}$ in [link] .

 $x$ $-3$ $-2$ $-1$ $0$ $1$ $2$ $3$ ${2}^{x}=y$ $\frac{1}{8}$ $\frac{1}{4}$ $\frac{1}{2}$ $1$ $2$ $4$ $8$ ${\mathrm{log}}_{2}\left(y\right)=x$ $-3$ $-2$ $-1$ $0$ $1$ $2$ $3$

Using the inputs and outputs from [link] , we can build another table to observe the relationship between points on the graphs of the inverse functions $\text{\hspace{0.17em}}f\left(x\right)={2}^{x}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}g\left(x\right)={\mathrm{log}}_{2}\left(x\right).\text{\hspace{0.17em}}$ See [link] .

 $f\left(x\right)={2}^{x}$ $\left(-3,\frac{1}{8}\right)$ $\left(-2,\frac{1}{4}\right)$ $\left(-1,\frac{1}{2}\right)$ $\left(0,1\right)$ $\left(1,2\right)$ $\left(2,4\right)$ $\left(3,8\right)$ $g\left(x\right)={\mathrm{log}}_{2}\left(x\right)$ $\left(\frac{1}{8},-3\right)$ $\left(\frac{1}{4},-2\right)$ $\left(\frac{1}{2},-1\right)$ $\left(1,0\right)$ $\left(2,1\right)$ $\left(4,2\right)$ $\left(8,3\right)$

As we’d expect, the x - and y -coordinates are reversed for the inverse functions. [link] shows the graph of $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}g.$

Observe the following from the graph:

• $f\left(x\right)={2}^{x}\text{\hspace{0.17em}}$ has a y -intercept at $\text{\hspace{0.17em}}\left(0,1\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}g\left(x\right)={\mathrm{log}}_{2}\left(x\right)\text{\hspace{0.17em}}$ has an x - intercept at $\text{\hspace{0.17em}}\left(1,0\right).$
• The domain of $\text{\hspace{0.17em}}f\left(x\right)={2}^{x},$ $\left(-\infty ,\infty \right),$ is the same as the range of $\text{\hspace{0.17em}}g\left(x\right)={\mathrm{log}}_{2}\left(x\right).$
• The range of $\text{\hspace{0.17em}}f\left(x\right)={2}^{x},$ $\left(0,\infty \right),$ is the same as the domain of $\text{\hspace{0.17em}}g\left(x\right)={\mathrm{log}}_{2}\left(x\right).$

## Characteristics of the graph of the parent function, f ( x ) = log b ( x )

For any real number $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ and constant $\text{\hspace{0.17em}}b>0,$ $b\ne 1,$ we can see the following characteristics in the graph of $\text{\hspace{0.17em}}f\left(x\right)={\mathrm{log}}_{b}\left(x\right):$

• one-to-one function
• vertical asymptote: $\text{\hspace{0.17em}}x=0$
• domain: $\text{\hspace{0.17em}}\left(0,\infty \right)$
• range: $\text{\hspace{0.17em}}\left(-\infty ,\infty \right)$
• x- intercept: $\text{\hspace{0.17em}}\left(1,0\right)\text{\hspace{0.17em}}$ and key point $\left(b,1\right)$
• y -intercept: none
• increasing if $\text{\hspace{0.17em}}b>1$
• decreasing if $\text{\hspace{0.17em}}0

[link] shows how changing the base $\text{\hspace{0.17em}}b\text{\hspace{0.17em}}$ in $\text{\hspace{0.17em}}f\left(x\right)={\mathrm{log}}_{b}\left(x\right)\text{\hspace{0.17em}}$ can affect the graphs. Observe that the graphs compress vertically as the value of the base increases. ( Note: recall that the function $\text{\hspace{0.17em}}\mathrm{ln}\left(x\right)\text{\hspace{0.17em}}$ has base $\text{\hspace{0.17em}}e\approx \text{2}.\text{718.)}$

hii
Amit
how are you
Dorbor
Find the possible value of 8.5 using moivre's theorem
which of these functions is not uniformly cintinuous on (0, 1)? sinx
which of these functions is not uniformly continuous on 0,1
solve this equation by completing the square 3x-4x-7=0
X=7
Muustapha
=7
mantu
x=7
mantu
3x-4x-7=0 -x=7 x=-7
Kr
x=-7
mantu
9x-16x-49=0 -7x=49 -x=7 x=7
mantu
what's the formula
Modress
-x=7
Modress
new member
siame
what is trigonometry
deals with circles, angles, and triangles. Usually in the form of Soh cah toa or sine, cosine, and tangent
Thomas
solve for me this equational y=2-x
what are you solving for
Alex
solve x
Rubben
you would move everything to the other side leaving x by itself. subtract 2 and divide -1.
Nikki
then I got x=-2
Rubben
it will b -y+2=x
Alex
goodness. I'm sorry. I will let Alex take the wheel.
Nikki
ouky thanks braa
Rubben
I think he drive me safe
Rubben
how to get 8 trigonometric function of tanA=0.5, given SinA=5/13? Can you help me?m
More example of algebra and trigo
What is Indices
If one side only of a triangle is given is it possible to solve for the unkown two sides?
cool
Rubben
kya
Khushnama
please I need help in maths
Okey tell me, what's your problem is?
Navin
the least possible degree ?
(1+cosA)(1-cosA)=sin^2A
good
Neha
why I'm sending you solved question
Mirza
Teach me abt the echelon method
Khamis
exact value of cos(π/3-π/4)