Logarithmic Functions

Graphing Logarithmic Functions

The graph of a logarithmic function has a vertical asymptote and is defined on only one side of the asymptote.
The graph of a logarithmic function is a curve with a vertical asymptote. An asymptote is a line that a graph approaches as one of the variables approaches infinity or negative infinity. The yy-axis is the asymptote for a graph of a logarithmic function in the form:
f(x)=logbxf(x)=\log_b{x}
It also has an xx-intercept of (1,0)(1, 0). If b>1b>1, the graph will also contain the point (b,1)(b, 1). The domain is x>0x>0, and the range is all real numbers. Using interval notation, the domain is (0,)(0, \infty) and the range is (,)(-\infty, \infty).
The graph of a logarithmic function is a curve with a vertical asymptote. Graphs of the logarithmic functions for logarithms with bases 2 and 10, as well as the natural logarithm function, each have a vertical asymptote at x=0x=0. Each graph also contains the point (1,0)(1, 0).

Evaluating Logarithmic Functions

Logarithmic functions can be evaluated to plot points on a graph.
As with all functions, each input of a logarithmic function is associated with a single output. When evaluating a logarithmic function to generate a set of ordered pairs, it can be helpful to choose values of xx that will result in integer values of f(x)f(x). Then plot the ordered pairs and connect them with a smooth curve to graph the function.
Step-By-Step Example
Graphing a Logarithmic Function
Graph the logarithmic function:
f(x)=log5xf(x)=\log_5{x}
Step 1

Analyze the equation of the function.

f(x)=log5xf(x)=\log_5{x}

The base of the logarithm is 5. So, the values of xx represent powers of 5. The values of f(x)f(x) represent the exponents of those powers.

Step 2

Choose several values of xx that are whole-number powers of 5.

Since 50=15^0=1, start with 1. Then keep multiplying by 5 to generate other powers of 5.

xx
Explanation
11
50=15^{0}=1
55
51=55^{1}=5
2525
52=255^{2}=25
125125
53=1255^{3}=125
625625
54=6255^{4}=625
Step 3

Use the relationship between logarithms and exponents to identify the corresponding values of f(x)f(x).

xx
f(x)=log5xf(x)=\log_5{x}
Explanation
11
00
The logarithmic form of 50=15^{0}=1 is log51=0\log_{5}1=0.
55
11
The logarithmic form of 51=55^{1}=5 is log55=1\log_{5}5=1.
2525
22
The logarithmic form of 52=255^{2}=25 is log525=2\log_{5}25=2.
125125
33
The logarithmic form of 53=1255^{3}=125 is log5125=3\log_{5}125=3.
625625
44
The logarithmic form of 54=6255^{4}=625 is log5625=4\log_{5}625=4.
Solution
Plot the ordered pairs, and connect them with a smooth curve. The function has the form:
f(x)=logbxf(x)=\log_b{x}
So, the graph has a vertical asymptote at x=0x=0.

A parent function is a function of a certain type that has the simplest algebraic rule. The parent logarithmic function using a common logarithm, or base 10, is:
f(x)=logxf(x)=\log{x}
The parent logarithmic function using a natural logarithm, or base ee, is:
f(x)=lnxf(x)=\ln{x}
Graphing the parent functions using the log and ln functions on a calculator makes them easier to identify.
The graphs of both f(x)=logxf(x)=\log{x} and f(x)=lnxf(x)=\ln{x} contain the point (1,0)(1, 0) and have the yy-axis as an asymptote. The curve for f(x)=logxf(x)=\log{x} contains the point (10,1)(10, 1). The curve for f(x)=lnxf(x)=\ln{x} contains the point (e,1)(e, 1).

Analyzing the Base of a Logarithmic Function

For a logarithmic function of the form f(x)=logbxf(x)=\log_b{x}, the value of bb determines whether the function is increasing or decreasing.
If b>1 b > 1, the graph of a logarithmic function of the form f(x)=logbxf(x)=\log_b{x} is always increasing as values of xx increase. If 0<b<10 < b < 1 , the graph is always decreasing as values of xx increase.
The graphs of f(x)=logbxf(x)=\log_b{x} and f(x)=log1bxf(x)=\log_{\footnotesize{\frac{1}{b}}}{x} are shown, where b>1b>1. The value of f(x)=logbxf(x)=\log_b{x} is increasing for x>0x>0, and the graph of f(x)=log1bxf(x)=\log_{\footnotesize{\frac{1}{b}}}{x} is decreasing for x>0x>0.
The reciprocal of a nonzero number bb is 1b\frac{1}{b}. The bases of these logarithmic functions are bb and 1b\frac{1}{b}:
f(x)=logbxf(x)=\log_b{x}\;\;\;\;\;
So, the bases are reciprocals. The graphs of logarithmic functions with reciprocal bases are reflections of each other across the xx-axis.
Step-By-Step Example
Graphing Logarithms with Reciprocal Bases
Use the properties of logarithmic graphs to graph the two functions:
f(x)=log2xf(x)=log12xf(x)=\log_2{x}\;\;\;\;\;f(x)=\log_{\footnotesize{\frac{1}{2}}}{x}
Step 1

Evaluate the first function to identify several ordered pairs on the graph.

xx
f(x)=log2xf(x)=\log_{2}x
Explanation
0.250.25
2-2
The logarithmic form of 22=0.252^{-2}=0.25 is log20.25=2\log_{2}0.25=-2.
0.50.5
1-1
The logarithmic form of 21=0.52^{-1}=0.5 is log20.5=1\log_{2}0.5=-1.
22
11
The logarithmic form of 21=22^{1}=2 is log22=1\log_{2}2=1.
44
22
The logarithmic form of 22=42^{2}=4 is log24=2\log_2{4}=2.
Step 2
Use the ordered pairs from the first function to generate ordered pairs for the second function:
f(x)=log12xf(x)=\log_{\footnotesize{\frac{1}{2}}}{x}
The functions are reflections of each other across the xx-axis because their bases are reciprocals. To reflect an ordered pair across the xx-axis, keep the xx-coordinate the same, but use the opposite yy-coordinate.
xx
f(x)=log12xf(x)=\log_{\footnotesize{\frac{1}{2}}}{x}
0.250.25
22
0.50.5
11
22
1-1
44
2-2
Solution

Determine the xx-intercept and the asymptote.

A logarithmic function in the form f(x)=logbxf(x)=log_{b}x has an xx-intercept of (1,0)(1, 0) and the yy-axis as the vertical asymptote. Therefore, the logarithmic function has an xx-intercept at (1,0)(1, 0) with the yy-axis as the vertical asymptote.

Plot the xx-intercept and the ordered pairs. Connect them with continuous curves.
The first function, f(x)=log2xf(x)=\log_{2}x is increasing.

The second function, f(x)=log12xf(x)=\log_{\footnotesize{\frac{1}{2}}}x is decreasing.

Transformations of Logarithmic Functions

Transformations can be used to graph logarithmic functions.
Transformations of logarithmic graphs are similar to transformations of other parent functions. For example, a logarithmic parent function takes on the form:
f(x)=logbxf(x)=\log_b{x}
It can be translated vertically kk units and horizontally hh units. It can also be stretched or compressed by a factor of aa. In addition, it can be reflected across the xx-axis or yy-axis.

Translations of Logarithmic Functions

Vertical Translations Horizontal Translations
For the graph of f(x)=logbxf(x)=\log_b{x} and k>0k>0:
  • The graph of f(x)+k=logbx+kf(x)+k=\log_b{x}+k is translated up by kk units.
  • The graph of f(x)k=logbxkf(x)-k=\log_b{x}-k is translated down by kk units.
For the graph of f(x)=logbxf(x)=\log_b{x} and h>0h>0:
  • The graph of f(xh)=logb(xh)f(x-h)=\log_b{(x-h)} is translated right by hh units.
  • The graph of f(x+h)=logb(x+h)f(x+h)=\log_b{(x+h)} is translated left by hh units.

Stretches, Compressions, and Reflections of Logarithmic Functions

Stretches and Compressions Reflections
For the graph of f(x)=logbxf(x)=\log_b{x} and a>0a>0:
  • The graph of af(x)=alogbxaf(x)=a\cdot \log_b{x} is a vertical stretch of the graph of f(x)f(x) by a factor of aa if a>1a>1.
  • The graph of af(x)=alogbxaf(x)=a\cdot \log_b{x} is a vertical compression of the graph of f(x)f(x) by a factor of aa if 0<a<10 \lt a \lt 1.
For the graph of f(x)=logbxf(x)=\log_b{x}:
  • The graph of f(x)=logbx-f(x)=-\log_b{x} is a reflection of the graph of f(x)f(x) across the xx-axis.
  • The graph of f(x)=logb(x)f(-x)=\log_b{(-x)} is a reflection of the graph of f(x)f(x) across the yy-axis.

As an example, examine the functions:
f(x)=log2(x+1)f(x)=log2xf(x)=log2x3f(x)=\log_2{(x+1)}\hspace{20pt}f(x)=-\log_2{x}\hspace{20pt}f(x)=\log_2{x}-3
All of them have the same parent function:
f(x)=log2xf(x)=\log_2{x}
The way in which the parent function is transformed depends on the function rules. The rule for the first function indicates a horizontal translation of 1 unit to the left:
f(x)=log2(x+1)f(x)=\log_2{(x{\color{#c42126}{+1}})}
The rule for the second function indicates a reflection over the xx-axis:
f(x)=log2xf(x)={\color{#c42126}{-}}\log_2{x}
The rule for the third function indicates a vertical translation of 3 units down:
f(x)=log2x3f(x)=\log_2{x}{\color{#c42126}{-3}}

Transformations of y=log2xy=\log_2{x}

f(x)=log2(x+1)f(x)=\log_2{(x+1)} f(x)=log2xf(x)=-\log_2{x} f(x)=log2x3f(x)=\log_2{x}-3
The graph shifts left 1 unit. The graph reflects across the xx-axis. The graph shifts down 3 units.

Step-By-Step Example
Graphing Logarithms Using Transformations
Graph the given logarithmic function:
f(x)=log2(x+1)3f(x)=-\!\log_2{(x+1)}-3
Step 1

Identify the parent function.

The base in the logarithm in the given function is 2. So, the parent function is the simplest logarithmic function with base 2.

The parent function is:
f(x)=log2xf(x)=\log_2{x}
Step 2
Determine whether there is a reflection by rewriting the given function:
f(x)=1log2(x+1)3f(x)=-1\cdot\log_2{(x+1)}-3
The logarithmic expression on the right side has a –1. Therefore, the graph of the parent function is reflected over the xx-axis. Graph the reflection of the parent function.
Solution

Determine whether there are any translations.

  • Since h=1h=1, there is a translation 1 unit to the left.
  • Since k=3k=-3, there is a translation 3 units down.
Graph the translations to produce the graph of the given function:
f(x)=log2(x+1)3f(x)=-\log_2{(x+1)}-3