# Graphs of Logarithmic Functions

## Graphing the inverse of exponential functions using transformations

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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 kerosene.

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?

### Graphing Logarithmic Functions

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) = log2x is the inverse of the function g(x) = 2x, 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) = 2x contains the point (1, 2), while the graph of f(x) = log2x contains the point (2, 1).

Also, note that while that the graph of g(x) = 2x is asymptotic to the x-axis, the graph of f(x) = log2x is asymptotic to the y-axis. 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) = log2x for values of x between 0 and 1.

If x = 1/2, then f(1/2) = log2(1/2) = -1 because 2-1 = 1/2
If x = 1/4, then f(1/4) = log2(1/4) = -2 because 2-2 = 1/4
If x = 1/8, then f(1/8) = log2(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 y-axis. 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) = log2x. The table below summarizes how we can use the graph of this function to graph other related functions.

Equation Relationship to f(x) = log2x Domain
g(x) = log2(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) = log2(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) = log2(x) + a for a > 0 Obtain a graph of g by shifting the graph of f up a units. x > 0
g(x) = log2(x) - a for a > 0 Obtain a graph of g by shifting the graph of f down a units. x > 0
g(x) = alog2(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) = -alog2(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 x-axis. x > 0
g(x) = log2(-x) Obtain a graph of g by reflecting the graph of f over the y-axis. x < 0

### Examples

#### Example 1

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!

#### Example 2

Graph the function f(x) = log4x and state the domain and range of the function.

The function f(x) = log4x is the inverse of the function g(x) = 4x. 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 = log4x
1/4
1 0
4 1
16 2

The graph is asymptotic to the y-axis, so the domain of f is the set of all real numbers that are greater than 0. We can write this as a set:{xR|x>0}\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 R\begin{align*}\mathbb R\end{align*}.

#### Example 3

Graph the following functions: f(x) = log2(x), g(x) = log2(x) + 3, and h(x) = log2(x + 3).

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 log2x , 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) = log2(-3 + 3) = log20 = 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 4

Graph the function y=log2(x3)\begin{align*}y = log_2 (x-3)\end{align*}.

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 20=1\begin{align*}2^0 = 1\end{align*} that means x3=1\begin{align*}x - 3 = 1\end{align*} and x=4\begin{align*}x = 4\end{align*}. That means that when y=0,x=4\begin{align*}y = 0, x = 4\end{align*}.

Since 21=2\begin{align*}2^1 = 2\end{align*} that means x3=2\begin{align*}x - 3 = 2\end{align*} and x=5\begin{align*}x = 5\end{align*}. That means that when y=1,x=5\begin{align*}y = 1, x = 5\end{align*}.

The graph looks like:

#### Example 5

Which of the following functions is graphed in the image below? a) y=log4x\begin{align*}y = log_4 x\end{align*}, b) y=log4x\begin{align*}y = -log_4 x\end{align*}, or c) y=log4x\begin{align*}y = log_4 -x\end{align*}?

All three functions are varieties of log4x\begin{align*}log_4 x\end{align*}, but the image shows the function reflected across the y\begin{align*}y\end{align*} axis, therefore c) y=log4x\begin{align*}y = log_4 -x\end{align*} is correct.

#### Example 6

Graph the function y=log3x\begin{align*}y = log_3 x\end{align*}.

To graph y=log3x\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:

### Review

Identify the domain and range, then sketch the graph.

1. y=log6(x1)5\begin{align*} y = log_{6} (x - 1) - 5\end{align*}
2. y=log5(x1)+3\begin{align*} y = log_{5}(x - 1) + 3\end{align*}
3. y=log4(3x+11)5\begin{align*} y = log_{4} (3x + 11) -5 \end{align*}
4. y=log6(3x+14)+1\begin{align*} y = log_ {6}(3x + 14) + 1\end{align*}
5. y=log5(2x+2)+5\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) f(x)=log10x\begin{align*}f(x) = -log_{10} x\end{align*}
b) f(x)=log10x\begin{align*}f(x) = log_{10} x\end{align*}
c) f(x)=log10(2x)\begin{align*}f(x) = -log_{10} (-2x)\end{align*}
d) f(x)=log10(3x)\begin{align*}f(x) = log_{10} (-3x)\end{align*}
e) f(x)=log10x+3\begin{align*}f(x) = log_{10} x+3\end{align*}

Graph the following logarithmic functions.

1. y=log10(2x)\begin{align*} y = log_{10} (2x)\end{align*}
2. y=log1/2(x+2)\begin{align*} y = log_{1/2} (x+2)\end{align*}
3. y=log3(2x+2)\begin{align*} y = log_{3} (2x + 2)\end{align*}
4. - 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) 14. (__ , __)
(-1, 1/2) 15. (__ , __)
(0, 1) 16. (__ , __)
(1, 2) 17. (__ , __)
(2, 4) 18. (__ , __)
(3, 8) 19. (__ , __)

Graph logarithmic functions, using the inverse of the related exponential function. Then graph the pair of functions on the same axes.

1. y=log3x\begin{align*} y = log_{3}x\end{align*}
2. y=log5x\begin{align*} y = log_ {5}x\end{align*}

To see the Review answers, open this PDF file and look for section 3.6.

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### Vocabulary Language: English

TermDefinition
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 $y=a \cdot b^{x-h}+k$.
log "log" is the shorthand term for 'the logarithm of', as in "$\log_b n$" means "the logarithm, base $b$, of $n$."
Logarithmic function Logarithmic functions are the inverses of exponential functions. Recall that $\log_b n=a$ is equivalent to $b^a=n$.
operation Operations are actions performed on variables, constants, or expressions. Common operations are addition, subtraction, multiplication, and division.