3.6: Graphs of Logarithmic Functions
Logarithmic functions are in many ways similar to exponential functions, and can be applied in similar situations.
Log functions can be used to model radioactive decay (to estimate the age of fossils, for instance), or estimate the discharge of electricity from a capacitor. Logarithmic functions can be graphed to allow quick and reasonably accurate estimation of the useable area of a WiFi network, or estimate the minimum length of a filtering pipe in order to have a specific purity of kerosine.
As you work with the graphs of log functions in this lesson, you will notice that they bear a resemblance to another family of graphs which you have explored already. At the end of the lesson, we will revisit this fact.
Can you identify what other family the log functions resemble, and explain in your own words why that is the case before the review?
Watch This
Embedded Video:
 James Sousa: Ex: Graph an Exponential Function and Logarithmic Function
Guidance
In a previous lesson, log functions were identified as the inverses of exponential functions, in this lesson we explore that fact visually through the graphs of logarithmic functions.
As you can see below, because the function f(x)=log_{2}x is the inverse of the function g(x)=2^{x}, the graphs of these functions are reflections over the line y = x.
We can verify that the functions are inverses by looking at the graph. For example, the graph of g(x)=2^{x} contains the point (1, 2), while the graph of f(x)=log_{2}x contains the point (2, 1).
Also, note that while that the graph of g(x)=2^{x} is asymptotic to the xaxis, the graph of f(x)=log_{2}x is asymptotic to the yaxis. This behavior of the graphs gives us a visual interpretation of the restricted range of g and the restricted domain of f.
When graphing log functions, it is important to consider x values across the domain of the function. In particular, we should look at the behavior of the graph as it gets closer and closer to the asymptote. Consider f(x)=log_{2}x for values of x between 0 and 1.
 If x=1/2, then f(1/2)=log_{2}(1/2)=1 because 2^{1}=1/2
 If x=1/4, then f(1/4)=log_2(1/4)=2 because 2^{2}=1/4
 If x=1/8}, then f(1/8)=log_{2}(1/8)=3 because 2^{3}=1/8
From these values you can see that if we choose x values that are closer and closer to 0, the y values decrease (heading towards \begin{align*} \infty \end{align*}!). In terms of the graph, these values show us that the graph gets closer and closer to the yaxis. Formally we say that the vertical asymptote of the graph is x = 0.
Graphing Logarithmic Functions Using Transformations
Consider again the log function f(x)=log_{2}x. The table below summarizes how we can use the graph of this function to graph other related functions.
Equation  Relationship to f(x)=log_{2}x  Domain 

g(x)=log_{2}(x  a), for a > 0  Obtain a graph of g by shifting the graph of f a units to the right.  x > a 
g(x) = log_{2}(x+a) for a > 0  Obtain a graph of g by shifting the graph of f a units to the left.  x > a 
g(x)=log_{2}(x) + a for a > 0  Obtain a graph of g by shifting the graph of f up a units.  x > 0 
g(x)=log_{2}(x)a for a > 0  Obtain a graph of g by shifting the graph of f down a units.  x > 0 
g(x)=alog_{2}(x) for a > 0  Obtain a graph of g by vertically stretching the graph of f by a factor of a.  x > 0 
g(x)=alog_{2}(x) , for a > 0  Obtain a graph of g by vertically stretching the graph of f by a factor of a, and by reflecting the graph over the xaxis.  x > 0 
g(x)=log_{2}(x)  Obtain a graph of g by reflecting the graph of f over the yaxis.  x < 0 
Example A
Graph the function f(x)=log_{4}x and state the domain and range of the function.
Solution:
The function f(x)=log_{4}x is the inverse of the function g(x) = 4^{x}. We can sketch a graph of f(x) by evaluating the function for several values of x, or by reflecting the graph of g over the line y = x.
 If we choose to plot points, it is helpful to organize the points in a table:
x  y=log_{4}x 

1/4  
1  0 
4  1 
16  2 
The graph is asymptotic to the yaxis, so the domain of f is the set of all real numbers that are greater than 0. We can write this as a set:\begin{align*}\left \{{x \in \mathbb R  x>0} \right \}\end{align*} . While the graph might look as if it has a horizontal asymptote, it does in fact continue to rise. The range is \begin{align*}\mathbb R\end{align*}.
Example B
Graph the functions:
 f(x) = log_{2}(x)
 g(x) = log_{2}(x) + 3
 h(x) = log_{2}(x + 3)
Solution:
The graph below shows these three functions together:
Notice that the location of the 3 in the equation makes a difference! When the 3 is added to log_{2}x , the shift is vertical. When the 3 is added to the x, the shift is horizontal. It is also important to remember that adding 3 to the x is a horizontal shift to the left. This makes sense if you consider the function value when x = 3:
 h(3)=log_{2}(3 + 3)=log_{2}0 = undefined
This is the vertical asymptote!
Note that in order to graph these functions, we evaluated them by investigating specific values of x. If we want to know what the x value is for a particular y value, we need to solve a logarithmic equation.
Example C
Graph the function \begin{align*}y = log_2 (x3)\end{align*}
Solution
Start by making a table:
x  y 

3  \begin{align*}\infty\end{align*} 
4  0 
5  1 
6  1.585 
7  2 
8  2.32 
Since \begin{align*}2^0 = 1\end{align*} that means \begin{align*}x  3 = 1\end{align*} and \begin{align*}x = 4\end{align*}
 That means that when \begin{align*}y = 0, x = 4\end{align*}
Since \begin{align*}2^1 = 2\end{align*} that means \begin{align*}x  3 = 2\end{align*} and \begin{align*}x = 5\end{align*}
 That means that when \begin{align*}y = 1, x = 5\end{align*}
The graph looks like:
Do you recall the question from the introduction to the lesson? Can you identify what other family the log functions resemble, and explain in your own words why that is the case? By now, I am sure you are quite familiar with the answer, it was repeated several times throughout the lesson. Logarithmic function graphs are the inverses of exponential function graphs, because the very definition of a logarithmic function it as the inverse of an exponential function! 

Vocabulary
Exponential functions are functions with the input variable (the x term) in the exponent.
Logarithmic functions are the inverse of exponential functions. Recall: log_{b}n = a is equivalent to b^{a}=n.
log is the shorthand term for 'the logarithm of', as in: "log_{b}n" = "the logarithm, base 'b', of 'n' ".
Guided Practice
1) Which of the following functions is graphed in the image below?
 a) \begin{align*}y = log_4 x\end{align*}
 b) \begin{align*}y = log_4 x\end{align*}
 c) \begin{align*}y = log_4 x\end{align*}
2) Graph the function \begin{align*}y = log_3 x\end{align*}
3) Transform the graph of \begin{align*}y = log_3 x\end{align*} from problem #2 into the graph of \begin{align*}y = log_3 (x +3)\end{align*}
Answers
1) All three functions are varieties of \begin{align*}log_4 x\end{align*}
 The image shows the function reflected across the \begin{align*}y\end{align*} axis, therefore:
 c) \begin{align*}y = log_4 x\end{align*} is correct.
2) To graph \begin{align*}y = log_3 x\end{align*} we will start with a table of values:
x  y 

3  1 
9  2 
27  3 
81  4 
Plotting those points and drawing a smooth curve between them gives:
3) The graph of \begin{align*}y = log_3 x\end{align*} above ran up the positive side of the yaxis to reach the xaxis.
The "+3" inside the parenthesis of \begin{align*}y = log_3 (x +3)\end{align*} means there is a shift of 3 to the left.
The image of \begin{align*}y = log_3 x\end{align*} shifted 3 units to the left looks like this:
Practice
Identify the domain and range, then sketch the graph.
 \begin{align*} y = log_{6} (x  1)  5\end{align*}
 \begin{align*} y = log_{5}(x  1) + 3\end{align*}
 \begin{align*} y = log_{4} (3x + 11) 5 \end{align*}
 \begin{align*} y = log_ {6}(3x + 14) + 1\end{align*}
 \begin{align*} y = log_ {5}(2x + 2) + 5\end{align*}
Look at the graphs below and identify the function that the graph represents from the functions listed below.
 a) \begin{align*}f(x) = log_{10} x\end{align*}
 b) \begin{align*}f(x) = log_{10} x\end{align*}
 c) \begin{align*}f(x) = log_{10} (2x)\end{align*}
 d) \begin{align*}f(x) = log_{10} (3x)\end{align*}
 e) \begin{align*}f(x) = log_{10} x+3\end{align*}
Graph the following logarithms.
 \begin{align*} y = log_{10} (2x)\end{align*}
 \begin{align*} y = log_{1/2} (x+2)\end{align*}
 \begin{align*} y = log_{3} (2x + 2)\end{align*}
  19. The table below shows the x and y values of the points on an exponential curve. Switch them and identify the corresponding coordinates of the points that would appear on the logarithmic curve. Can you identify the function?


Point on exponential curve Corresponding point on logarithmic curve (3, 1/8) (1/8, 3) (2, 1/4) (__ , __) (1, 1/2) (__ , __) (0, 1) (__ , __) (1, 2) (__ , __) (2, 4) (__ , __) (3, 8) (__ , __)

Graph logarithmic functions, using the inverse of the related exponential function. Then graph the pair of functions on the same axes.
 \begin{align*} y = log_{3}x\end{align*}
 \begin{align*} y = log_ {5}x\end{align*}
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Term  Definition 

Asymptotes  An asymptote is a line on the graph of a function representing a value toward which the function may approach, but does not reach (with certain exceptions). 
Exponential Function  An exponential function is a function whose variable is in the exponent. The general form is . 
log  "log" is the shorthand term for 'the logarithm of', as in "" means "the logarithm, base , of ." 
Logarithmic function  Logarithmic functions are the inverses of exponential functions. Recall that is equivalent to . 
operation  Operations are actions performed on variables, constants, or expressions. Common operations are addition, subtraction, multiplication, and division. 
Image Attributions
Here you will explore the graphs of logarithmic functions, and compare them to the graphs of exponential functions. You will also see how the common rules for graphing using transformations apply to log graphs.