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# Graphs of Logarithmic Functions

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Graphing Logarithmic Functions
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Your math homework assignment is to find out which quadrants the graph of the function $f(x)= 4\ln(x + 3)$ 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?

### Guidance

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 $f(x)=\log_b(x-h)+k$ 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 up to the right. If $0 , the graph moves down to the right.

#### Example A

Graph $y=\log_3(x-4)$ . State the domain and range.

Solution:

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 “easy” points, such as (5, 0), (7, 1), and (13, 2) and connect.

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

#### Example B

Is (16, 1) on $y=\log (x-6)$ ?

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

$1 &= \log(16-6) \\1 &= \log 10 \\1 &= 1$

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

#### Example C

Graph $f(x)=2 \ln(x+1)$ .

Solution: To graph a natural log, we need to use a graphing calculator. Press $Y=$ and enter in the function, $Y=2 \ln(x+1)$ , GRAPH .

Intro Problem Revisit the vertical asymptote of the function $f(x)= 4\ln(x + 3)$ is $x = -3$ since x will approach $-3$ but never quite reach it, x can assume some negative values. Hence, the function will fall in Quadrants II and III. Therefore, you are correct and your friend is wrong.

### Guided Practice

1. Graph $y=\log_{\frac{1}{4}} x+2$ in an appropriate window.

2. Graph $y=-\log x$ using a graphing calculator. Find the domain and range.

3. Is (-2, 1) on the graph of $f(x)=\log_{\frac{1}{2}} (x+4)$ ?

#### Answers

1. 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 . The keystrokes would be:

$Y= \frac{\log(x)}{\log \left ( \frac{1}{4} \right )}+2$ , GRAPH

To see a table of values, press $2^{nd} \rightarrow$ GRAPH .

2. The keystrokes are $Y=-\log(x)$ , GRAPH .

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

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

$1 &= \log_{\frac{1}{2}} (-2+4) \\1 &= \log_{\frac{1}{2}} 2 \rightarrow \frac{1}{2}^x=2 \\1 & \ne -1$

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

### Practice

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

1. $y=\log_5 x$
2. $y=\log_2(x+1)$
3. $y=\log(x)-4$
4. $y=\log_{\frac{1}{3}}(x-1)+3$
5. $y=-\log_{\frac{1}{2}}(x+3)-5$
6. $y=\log_4(2-x)+2$

Graph the following logarithmic functions using your graphing calculator.

1. $y=\ln (x+6)-1$
2. $y=-\ln (x-1)+2$
3. $y=\log(1-x)+3$
4. $y=\log(x+2)-4$
5. How would you graph $y=\log_4 x$ on the graphing calculator? Graph the function.
6. Graph $y=\log_{\frac{3}{4}}x$ on the graphing calculator.
7. Is (3, 8) on the graph of $y=\log_3 (2x-3)+7$ ?
8. Is (9, -2) on the graph of $y=\log_{\frac{1}{4}} (x-5)$ ?
9. Is (4, 5) on the graph of $y=5 \log_2 (8-x)$ ?

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