# 3.4.1: Graphs of Logarithmic Functions

- Page ID
- 14376

# Graphing Logarithmic Functions

Your math homework assignment is to find out which quadrants the graph of the function f(x)=4ln(x+3) falls in. On the way home, your best friend tells you, "This is the easiest homework assignment ever! All logarithmic functions fall in Quadrants I and IV." You're not so sure, so you go home and graph the function as instructed. Your graph falls in Quadrant I as your friend thought, but instead of Quadrant IV, it also falls in Quadrants II and III. Which one of you is correct?

# Graphing Logarithmic Functions

Now that we are more comfortable with using these functions as inverses, let’s use this idea to graph a logarithmic function. Recall that functions are inverses of each other when they are mirror images over the line y=x. Therefore, if we reflect y=b^{x} over y=x, then we will get the graph of y=log_{b}x.

Recall that an exponential function has a horizontal asymptote. Because the logarithm is its inverse, it will have a * vertical* asymptote. The general form of a logarithmic function is f(x)=a log

_{b}(x−h)+k and the vertical asymptote is x=h. The domain is x>h and the range is all real numbers. Lastly, if b>1, the graph moves

*to the right. If 0<b<1, the graph moves*

*up**to the right.*

*down*Let's graph y=log_{3}(x−4) and state the domain and range.

To graph a logarithmic function without a calculator, start by drawing the vertical asymptote, at x=4. We know the graph is going to have the general shape of the first function above. Plot a few points, such as (5, 0), (7, 1), and (13, 2) and connect.

The domain is x>4 and the range is all real numbers.

Now, let's determine if (16, 1) is on y=log(x−6).

Plug in the point to the equation to see if it holds true.

\(\ \begin{array}{l}

1=\log (16-6) \\

1=\log 10 \\

1=1

\end{array}\)

Yes, this is true, so (16, 1) is on the graph.

Finally, let's graph f(x)=2ln(x+1).

To graph a natural log, we can use a graphing calculator. Press Y= and enter in the function, Y=2ln(x+1), ** GRAPH**.

# Examples

Example 1

Earlier, you were asked to determine if your friend was correct.

**Solution**

The vertical asymptote of the function f(x)=4ln(x+3) is x=−3. Since * x* will approach −3 but never quite reach it,

*can assume some negative values. Hence, the function will fall in Quadrants II and III. Therefore, you are correct and your friend is wrong.*

*x*Example 2

Graph \(\ y=\log _{\frac{1}{4}} x+2\) in an appropriate window.

**Solution**

First, there is a vertical asymptote at x=0. Now, determine a few easy points, points where the log is easy to find; such as (1, 2), (4, 1), (8, 0.5), and (16, 0).

To graph a logarithmic function using a TI-83/84, enter the function into Y= and use the Change of Base Formula: \(\ \log _{a} x=\frac{\log _{b} x}{\log _{b} a}\). The keystrokes would be: \(\ Y=\frac{\log (x)}{\log \left(\frac{1}{4}\right)}+2\), **GRAPH**

To see a table of values, press 2^{nd}→ ** GRAPH**.

Example 3

Graph y=−logx using a graphing calculator. Find the domain and range.

**Solution**

The keystrokes are Y=−log(x), ** GRAPH**.

The domain is x>0 and the range is all real numbers.

Example 4

Is (-2, 1) on the graph of \(\ f(x)=\log _{\frac{1}{2}}(x+4)\)?

**Solution**

Plug (-2, 1) into \(\ f(x)=\log _{\frac{1}{2}}(x+4)\) to see if the equation holds true.

\(\ \begin{array}{l}

1=\log _{\frac{1}{2}}(-2+4) \\

1=\log _{\frac{1}{2}} 2 \\

1 \neq-1

\end{array}\)

Therefore, (-2, 1) is not on the graph. However, (-2, -1) is.

# Review

Graph the following logarithmic functions without using a calculator. State the equation of the asymptote, the domain and the range of each function.

- \(\ y=\log _{5} x\)
- \(\ y=\log _{2}(x+1)\)
- \(\ y=\log (x)-4\)
- \(\ y=\log _{\frac{1}{3}}(x-1)+3\)
- \(\ y=-\log _{\frac{1}{2}}(x+3)-5\)
- \(\ y=\log _4(2-x)+2\)

Graph the following logarithmic functions using your graphing calculator.

- \(\ y=\ln(x+6)-1\)
- \(\ y=-\ln(x-1)+2\)
- \(\ y=\log(1-x)+3\)
- \(\ y=\log(x+2)-4\)
- How would you graph \(\ y=\log_4x\) on the graphing calculator? Graph the function.
- Graph \(\ y=\log_{\frac{3}{4}}x\) on the graphing calculator.
- Is (3, 8) on the graph of \(\ y=\log_3(2x-3)+7\)?
- Is (9, -2) on the graph of \(\ y=\log_{\frac{1}{4}}(x-5)\)?
- Is (4, 5) on the graph of \(\ y=5\log_2(8-x)\)?

# Vocabulary

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). |

operation |
Operations are actions performed on variables, constants, or expressions. Common operations are addition, subtraction, multiplication, and division. |

# Image Attributions

- [Figure 1]

CK-12**Credit:**

CK-12**Source:** - [Figure 2]

CK-12**Credit:**

CK-12**Source:** - [Figure 3]

CK-12 Foundation**Credit:**