<< Chapter < Page Chapter >> Page >

Given a logarithmic function, identify the domain.

  1. Set up an inequality showing the argument greater than zero.
  2. Solve for x .
  3. Write the domain in interval notation.

Identifying the domain of a logarithmic shift

What is the domain of f ( x ) = log 2 ( x + 3 ) ?

The logarithmic function is defined only when the input is positive, so this function is defined when x + 3 > 0. Solving this inequality,

x + 3 > 0 The input must be positive . x > 3 Subtract 3 .

The domain of f ( x ) = log 2 ( x + 3 ) is ( 3 , ) .

Got questions? Get instant answers now!
Got questions? Get instant answers now!

What is the domain of f ( x ) = log 5 ( x 2 ) + 1 ?

( 2 , )

Got questions? Get instant answers now!

Identifying the domain of a logarithmic shift and reflection

What is the domain of f ( x ) = log ( 5 2 x ) ?

The logarithmic function is defined only when the input is positive, so this function is defined when 5 2 x > 0 . Solving this inequality,

5 2 x > 0 The input must be positive . 2 x > 5 Subtract  5. x < 5 2 Divide by  2  and switch the inequality .

The domain of f ( x ) = log ( 5 2 x ) is ( , 5 2 ) .

Got questions? Get instant answers now!
Got questions? Get instant answers now!

What is the domain of f ( x ) = log ( x 5 ) + 2 ?

( 5 , )

Got questions? Get instant answers now!

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 y = log b ( x ) along with all its transformations: shifts, stretches, compressions, and reflections.

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

x 3 2 1 0 1 2 3
2 x = y 1 8 1 4 1 2 1 2 4 8
log 2 ( y ) = 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 f ( x ) = 2 x and g ( x ) = log 2 ( x ) . See [link] .

f ( x ) = 2 x ( 3 , 1 8 ) ( 2 , 1 4 ) ( 1 , 1 2 ) ( 0 , 1 ) ( 1 , 2 ) ( 2 , 4 ) ( 3 , 8 )
g ( x ) = log 2 ( x ) ( 1 8 , 3 ) ( 1 4 , 2 ) ( 1 2 , 1 ) ( 1 , 0 ) ( 2 , 1 ) ( 4 , 2 ) ( 8 , 3 )

As we’d expect, the x - and y -coordinates are reversed for the inverse functions. [link] shows the graph of f and g .

Graph of two functions, f(x)=2^x and g(x)=log_2(x), with the line y=x denoting the axis of symmetry.
Notice that the graphs of f ( x ) = 2 x and g ( x ) = log 2 ( x ) are reflections about the line y = x .

Observe the following from the graph:

  • f ( x ) = 2 x has a y -intercept at ( 0 , 1 ) and g ( x ) = log 2 ( x ) has an x - intercept at ( 1 , 0 ) .
  • The domain of f ( x ) = 2 x , ( , ) , is the same as the range of g ( x ) = log 2 ( x ) .
  • The range of f ( x ) = 2 x , ( 0 , ) , is the same as the domain of g ( x ) = log 2 ( x ) .

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

For any real number x and constant b > 0 , b 1 , we can see the following characteristics in the graph of f ( x ) = log b ( x ) :

  • one-to-one function
  • vertical asymptote: x = 0
  • domain: ( 0 , )
  • range: ( , )
  • x- intercept: ( 1 , 0 ) and key point ( b , 1 )
  • y -intercept: none
  • increasing if b > 1
  • decreasing if 0 < b < 1

See [link] .

Two graphs of the function f(x)=log_b(x) with points (1,0) and (b, 1). The first graph shows the line when b>1, and the second graph shows the line when 0<b<1.

[link] shows how changing the base b in f ( x ) = log b ( x ) can affect the graphs. Observe that the graphs compress vertically as the value of the base increases. ( Note: recall that the function ln ( x ) has base e 2 . 718.)

Graph of three equations: y=log_2(x) in blue, y=ln(x) in orange, and y=log(x) in red. The y-axis is the asymptote.
The graphs of three logarithmic functions with different bases, all greater than 1.

Questions & Answers

give me an example of a problem so that I can practice answering
Jenefa Reply
x³+y³+z³=42
Robert
dont forget the cube in each variable ;)
Robert
of she solves that, well ... then she has a lot of computational force under her command ....
Walter
what is a function?
CJ Reply
I want to learn about the law of exponent
Quera Reply
explain this
Hinderson Reply
what is functions?
Angel Reply
A mathematical relation such that every input has only one out.
Spiro
yes..it is a relationo of orders pairs of sets one or more input that leads to a exactly one output.
Mubita
Is a rule that assigns to each element X in a set A exactly one element, called F(x), in a set B.
RichieRich
If the plane intersects the cone (either above or below) horizontally, what figure will be created?
Feemark Reply
can you not take the square root of a negative number
Sharon Reply
No because a negative times a negative is a positive. No matter what you do you can never multiply the same number by itself and end with a negative
lurverkitten
Actually you can. you get what's called an Imaginary number denoted by i which is represented on the complex plane. The reply above would be correct if we were still confined to the "real" number line.
Liam
Suppose P= {-3,1,3} Q={-3,-2-1} and R= {-2,2,3}.what is the intersection
Elaine Reply
can I get some pretty basic questions
Ama Reply
In what way does set notation relate to function notation
Ama
is precalculus needed to take caculus
Amara Reply
It depends on what you already know. Just test yourself with some precalculus questions. If you find them easy, you're good to go.
Spiro
the solution doesn't seem right for this problem
Mars Reply
what is the domain of f(x)=x-4/x^2-2x-15 then
Conney Reply
x is different from -5&3
Seid
All real x except 5 and - 3
Spiro
***youtu.be/ESxOXfh2Poc
Loree
how to prroved cos⁴x-sin⁴x= cos²x-sin²x are equal
jeric Reply
Don't think that you can.
Elliott
By using some imaginary no.
Tanmay
how do you provided cos⁴x-sin⁴x = cos²x-sin²x are equal
jeric Reply
What are the question marks for?
Elliott
Someone should please solve it for me Add 2over ×+3 +y-4 over 5 simplify (×+a)with square root of two -×root 2 all over a multiply 1over ×-y{(×-y)(×+y)} over ×y
Abena Reply
For the first question, I got (3y-2)/15 Second one, I got Root 2 Third one, I got 1/(y to the fourth power) I dont if it's right cause I can barely understand the question.
Is under distribute property, inverse function, algebra and addition and multiplication function; so is a combined question
Abena

Get the best Precalculus course in your pocket!





Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Precalculus' conversation and receive update notifications?

Ask