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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 , ) .

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

( 2 , )

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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 ) .

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

( 5 , )

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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

how to solve the Identity ?
Barcenas Reply
what type of identity
Jeffrey
Confunction Identity
Barcenas
For each year t, the population of a forest of trees is represented by the function A(t) = 117(1.029)t. In a neighboring forest, the population of the same type of tree is represented by the function B(t) = 86(1.025)t.
Shakeena Reply
by how many trees did forest "A" have a greater number?
Shakeena
32.243
Kenard
how solve standard form of polar
Rhudy Reply
what is a complex number used for?
Drew Reply
It's just like any other number. The important thing to know is that they exist and can be used in computations like any number.
Steve
I would like to add that they are used in AC signal analysis for one thing
Scott
Good call Scott. Also radar signals I believe.
Steve
They are used in any profession where the phase of a waveform has to be accounted for in the calculations. Imagine two electrical signals in a wire that are out of phase by 90°. At some times they will interfere constructively, others destructively. Complex numbers simplify those equations
Tim
Is there any rule we can use to get the nth term ?
Anwar Reply
how do you get the (1.4427)^t in the carp problem?
Gabrielle Reply
A hedge is contrusted to be in the shape of hyperbola near a fountain at the center of yard.the hedge will follow the asymptotes y=x and y=-x and closest distance near the distance to the centre fountain at 5 yards find the eqution of the hyperbola
ayesha Reply
A doctor prescribes 125 milligrams of a therapeutic drug that decays by about 30% each hour. To the nearest hour, what is the half-life of the drug?
Sandra Reply
Find the domain of the function in interval or inequality notation f(x)=4-9x+3x^2
prince Reply
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Jessica Reply
Outside temperatures over the course of a day can be modeled as a sinusoidal function. Suppose the high temperature of ?105°F??105°F? occurs at 5PM and the average temperature for the day is ?85°F.??85°F.? Find the temperature, to the nearest degree, at 9AM.
Karlee Reply
if you have the amplitude and the period and the phase shift ho would you know where to start and where to end?
Jean Reply
rotation by 80 of (x^2/9)-(y^2/16)=1
Garrett Reply
thanks the domain is good but a i would like to get some other examples of how to find the range of a function
bashiir Reply
what is the standard form if the focus is at (0,2) ?
Lorejean Reply

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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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