Graphing Logarithmic Functions

Show your work for each problem.

1.
What is the domain of $y = \log_3 x$ ?
1. a) $x > 3$
2. b) $x \geq 0$
3. c) $x > 0$
4. d) $x < 0$
Answer: $x > 0$
$3^y$ is always positive, so $x = 3^y$ must be positive. The domain is $x > 0$ ; $x = 0$ is not included because $\log_3 0$ does not exist.
2.
The point $(8, y)$ lies on the graph of $y = \log_2 x$ . What is $y$ ?
Answer: 3
$2^3 = 8$ , so $\log_2 8 = 3$ and the point is $(8, 3)$ .
3.
Complete the table of values for $y = \log_3 x$ at $x = \frac{1}{3}, 1, 3, 9$ .
Answer: -1, 0, 1, 2
1. What is $y$ at $x = \frac{1}{3}$ ? -1
2. What is $y$ at $x = 1$ ? 0
3. What is $y$ at $x = 3$ ? 1
4. What is $y$ at $x = 9$ ? 2
4.
What is the vertical asymptote of $y = \log_2(x + 4)$ ?
1. a) $x = -4$
2. b) $x = 4$
3. c) $y = 0$
4. d) $y = -4$
Answer: $x = -4$
The argument $x + 4$ is $0$ at $x = -4$ . The graph of $y = \log_2 x$ moves 4 units left, so the asymptote is $x = -4$ .
5.
At which $x$ does the graph of $y = \log_5 x$ cross the $x$ -axis?
Answer: 1
$\log_5 x = 0$ means $x = 5^0 = 1$ . Every graph $y = \log_b x$ crosses the $x$ -axis at $(1, 0)$ .
6.
The point $(2, 9)$ lies on $y = 3^x$ . Find the matching point on $y = \log_3 x$ .
Answer: (9, 2)
1. What is the x-coordinate of the reflected point? 9
2. What is the y-coordinate of the reflected point? 2
3. Check: what is $\log_3 9$ ? 2
7.
Which function has a decreasing graph?
1. a) $y = \ln x$
2. b) $y = \log_5 x$
3. c) $y = \log_{10} x$
4. d) $y = \log_{1/3} x$
Answer: $y = \log_{1/3} x$
$\log_{1/3} x$ has the base $\frac{1}{3}$ , which lies between 0 and 1, so its graph decreases. The bases $5$ , $10$ and $e$ are greater than 1, so those graphs increase.
8.
The graph of $y = \log_3(x - 7) + 2$ has the vertical asymptote $x = c$ . What is $c$ ?
Answer: 7
The argument $x - 7$ is $0$ at $x = 7$ , so the asymptote is $x = 7$ . The $+2$ only shifts the graph up and does not change the asymptote.
9.
Graph $y = \log_2(x - 5)$ : find the asymptote and two key points.
Answer: $x = 5$ ; $(6, 0)$ , $(7, 1)$
1. The asymptote is $x = ?$ 5
2. For which $x$ is $y = 0$ ? 6
3. For which $x$ is $y = 1$ ? 7
10.
On the graph of $y = \log_{1/2} x$ , how much does $y$ change when $x$ goes from $\frac{1}{2}$ to $4$ ?
Answer: -3
1. What is $\log_{1/2} \frac{1}{2}$ ? 1
2. What is $\log_{1/2} 4$ ? -2
3. What is the change in $y$ (end minus start)? -3
11.
The graph of $y = \log_4 x$ is the reflection across the line $y = x$ of which graph?
1. a) $y = 4x$
2. b) $y = 4^x$
3. c) $y = x^4$
4. d) $y = \frac{x}{4}$
Answer: $y = 4^x$
$y = \log_4 x$ means $4^y = x$ , so it is the inverse of $y = 4^x$ . Inverse functions are reflections of each other across $y = x$ .
12.
What is $y$ on the graph of $y = \log_2(x + 1) - 3$ at $x = 7$ ?
Answer: 0
$y = \log_2 8 - 3 = 3 - 3 = 0$ . So $(7, 0)$ is the $x$ -intercept of this graph.
13.
The graph of $y = \log_2 x$ is shifted to $y = \log_2(x + 3) - 4$ . Where do the key points $(1, 0)$ and $(2, 1)$ go?
Answer: $(-2, -4)$ , $(-1, -3)$
1. New x-coordinate of $(1, 0)$ ? -2
2. New y-coordinate of $(1, 0)$ ? -4
3. New x-coordinate of $(2, 1)$ ? -1
4. New y-coordinate of $(2, 1)$ ? -3
14.
Which function has the vertical asymptote $x = 2$ and passes through $(3, -1)$ ?
1. a) $y = \log_2(x - 2) + 1$
2. b) $y = \log_2(x - 2) - 1$
3. c) $y = \log_2(x + 2) - 1$
4. d) $y = \log_2(x - 1) - 1$
Answer: $y = \log_2(x - 2) - 1$
Asymptote $x = 2$ needs the argument $x - 2$ , which leaves $\log_2(x - 2) - 1$ and $\log_2(x - 2) + 1$ . At $x = 3$ : $\log_2 1 - 1 = -1$ , but $\log_2 1 + 1 = 1$ . So the answer is $y = \log_2(x - 2) - 1$ .
15.
Find the $x$ -intercept of $y = \log_3(x - 1) - 2$ . Enter the value of $x$ .
Answer: 10
$\log_3(x - 1) = 2$ means $x - 1 = 3^2 = 9$ , so $x = 10$ . The $x$ -intercept is $(10, 0)$ .
16.
Graph $y = -\log_3(x + 2)$ : find the asymptote, the $x$ -intercept and two more points.
Answer: $x = -2$ ; $(-1, 0)$ , $(7, -2)$ , $\left(-\frac{5}{3}, 1\right)$
1. The asymptote is $x = ?$ -2
2. At which $x$ is $y = 0$ ? -1
3. What is $y$ at $x = 7$ ? -2
4. What is $y$ at $x = -\frac{5}{3}$ ? 1
17.
A graph $y = \log_b(x - h) + k$ has the asymptote $x = 1$ and passes through $(2, 3)$ and $(6, 4)$ . Find $h$ , $k$ and $b$ .
Answer: $y = \log_5(x - 1) + 3$
1. What is $h$ ? 1
2. What is $k$ ? 3
3. What is $b$ ? 5
18.
Study $y = \log_2(6 - 2x)$ : find the asymptote, one point and the $x$ -intercept.
Answer: $x = 3$ ; $(1, 2)$ , $\left(\frac{5}{2}, 0\right)$
1. Solve $6 - 2x = 0$ . The asymptote is $x = ?$ 3
2. What is $y$ at $x = 1$ ? 2
3. At which $x$ is $y = 0$ ? (Write a decimal.) 2.5