Graphs of Logarithmic Functions

Problem: Let \( f \) be a function given by \( f(x) = \log_2(x + 2) \).

1. Find the domain and range of \( f \).
2. Find the vertical asymptote of the graph of \( f \).
3. Find the \( x \)- and \( y \)-intercepts of the graph of \( f \) if there are any.
4. Sketch the graph of \( f \).

Solution:

1. Domain & Range: The domain of \( f \) is the set of all \( x \) values such that: \[ x + 2 > 0 \implies x > -2 \] In interval notation, the domain is \((-2, +\infty)\). The range of \( f \) is given by the interval: \[ (-\infty, +\infty) \]
2. Vertical Asymptote: Obtained by solving the equation: \[ x + 2 = 0 \implies x = -2 \] As \( x \) approaches \(-2\) from the right (\( x \to -2^+ \)), the function \( f(x) \) decreases without bound (\( \lim_{x \to -2^+} f(x) = -\infty \)).
3. Intercepts:
   * To find the \( x \)-intercept, solve \( \log_2(x + 2) = 0 \): \[ x + 2 = 2^0 = 1 \implies x = -1 \quad \text{i.e., } (-1, 0) \] * To find the \( y \)-intercept, evaluate \( f(0) \): \[ f(0) = \log_2(0 + 2) = \log_2(2) = 1 \quad \text{i.e., } (0, 1) \]
4. Sketching & Key Points:
   Evaluating additional points (e.g., \( x = -\frac{3}{2} \to \log_2(0.5) = -1 \), and \( x = 2 \to \log_2(4) = 2 \)):

   

Problem: Let \( f \) be a function defined by \( f(x) = \log_2(x + 3) \).

1. Find the domain and range of \( f \).
2. Find the vertical asymptote of the graph of \( f \).
3. Find the \( x \)- and \( y \)-intercepts of the graph of \( f \) if there are any.
4. Sketch the graph of \( f \).

Problem: Let \( f \) be a function given by \( f(x) = -3\ln(x - 4) \).

1. Find the domain and range of \( f \).
2. Find the vertical asymptote of the graph of \( f \).
3. Find the \( x \)- and \( y \)-intercepts of the graph of \( f \) if there are any.
4. Sketch the graph of \( f \).

Solution:

1. Domain & Range: Set \( x - 4 > 0 \), yielding domain \( x > 4 \) or \((4, +\infty)\). The range is \((-\infty, +\infty)\).
2. Vertical Asymptote: Solve \( x - 4 = 0 \implies x = 4 \). As \( x \) approaches \( 4 \) from the right, \( f(x) \) increases without bound.
3. Intercepts:
   * \( x \)-intercept: Solve \( -3\ln(x - 4) = 0 \implies \ln(x - 4) = 0 \implies x - 4 = e^0 = 1 \implies x = 5 \), giving \( (5, 0) \).
   * \( y \)-intercept: \( f(0) \) is undefined since \( x = 0 \) is not in the domain. There is no \( y \)-intercept.
4. Sketching & Key Points:
   Calculating extra reference points: \( f(4.5) \approx 2.08 \), \( f(8) \approx -4.16 \), \( f(14) \approx -6.91 \).

   

Problem: Let \( f \) be a function given by \( f(x) = 2\ln(x + 5) \).

1. Find the domain and range of \( f \).
2. Find the vertical asymptote of the graph of \( f \).
3. Find the \( x \)- and \( y \)-intercepts of the graph of \( f \) if there are any.
4. Sketch the graph of \( f \).

Problem: Let \( f \) be a function given by \( f(x) = 2\ln(|x|) \).

1. Find the domain and range of \( f \).
2. Find the vertical asymptote of the graph of \( f \).
3. Find the \( x \)- and \( y \)-intercepts of the graph of \( f \) if there are any.
4. Sketch the graph of \( f \).

Solution:

1. Domain & Range: The domain requires \( |x| > 0 \), meaning all real numbers except \( 0 \) (i.e., \((-\infty, 0) \cup (0, +\infty)\)). The range is \((-\infty, +\infty)\).
2. Vertical Asymptote: Solve \( |x| = 0 \implies x = 0 \). As \( x \) approaches \( 0 \) from either side, \( f(x) \) decreases without bound.
3. Intercepts & Symmetry:
   * Since \( f(-x) = 2\ln(|-x|) = 2\ln(|x|) = f(x) \), \( f \) is an even function symmetric with respect to the \( y \)-axis.
   * \( x \)-intercepts: Solve \( 2\ln(|x|) = 0 \implies |x| = 1 \implies x = \pm 1 \), giving \( (1, 0) \) and \((-1, 0)\).
   * \( y \)-intercept: Undefined (\( x = 0 \) is not in the domain).
4. Sketching & Key Points:
   Extra points: \( f(4) \approx 2.77 \), \( f(0.5) \approx -1.39 \) (and by symmetry, \( f(-4) = 2.77 \), \( f(-0.5) = -1.39 \)).

   

Problem: Let \( f \) be a function given by \( f(x) = -2\ln(x^2) \).

1. Find the domain and range of \( f \).
2. Find the vertical asymptote of the graph of \( f \).
3. Find the \( x \)- and \( y \)-intercepts of the graph of \( f \) if there are any.
4. Sketch the graph of \( f \).