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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.

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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