4.4 Graphs of Logarithmic Functions

Graphing Logarithmic Functions

Logarithmic functions are the inverses of exponential functions, and their graphs reflect that relationship. Understanding how to graph them, including transformations and key features, is essential for working with logarithmic models throughout the rest of this course.

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The Parent Function and Transformations

The parent function for logarithms is $f(x) = \log_b(x)$, where $b$ is the base of the logarithm ($b > 0, b \neq 1$). The two most common bases you'll see are 10 (written as $\log x$, the "common log") and $e$ (written as $\ln x$, the "natural log").

From the parent function, you can apply transformations to shift, stretch, compress, or reflect the graph:

• Vertical shift: $f(x) = \log_b(x) + k$ moves the graph up $k$ units if $k > 0$, or down $|k|$ units if $k < 0$.
• Horizontal shift: $f(x) = \log_b(x - h)$ moves the graph right $h$ units if $h > 0$, or left $|h|$ units if $h < 0$. This also shifts the vertical asymptote from $x = 0$ to $x = h$.
• Vertical stretch/compression: $f(x) = a \cdot \log_b(x)$ stretches the graph vertically by a factor of $|a|$ when $|a| > 1$, or compresses it when $0 < |a| < 1$.
  • If $a < 0$, the graph is also reflected over the x-axis.
• Horizontal stretch/compression: $f(x) = \log_b(c \cdot x)$ compresses the graph horizontally by a factor of $\frac{1}{c}$ when $c > 1$, or stretches it by a factor of $\frac{1}{c}$ when $0 < c < 1$.
  • If $c < 0$, the graph is reflected over the y-axis. Note that this changes the domain to $x < 0$.

A quick tip for horizontal shifts: the expression inside the log tells you the new domain restriction. Set the argument greater than zero and solve. For example, $f(x) = \log_2(x - 3)$ requires $x - 3 > 0$, so the domain is $x > 3$.

Domain and Range of Logarithms

The domain of the parent logarithmic function is all positive real numbers: $(0, \infty)$. The argument of the logarithm (the expression inside the parentheses) must always be strictly greater than zero. For transformed functions, you find the domain by solving for when the argument is positive.

The range is all real numbers: $(-\infty, \infty)$. No matter the base, a logarithmic function can output any real number. As $x$ approaches 0 from the right, the output drops toward $-\infty$, and as $x$ grows, the output increases toward $+\infty$.

Key Features of Logarithmic Graphs

Vertical asymptote: The parent function has a vertical asymptote at $x = 0$. The graph gets infinitely close to this line but never touches or crosses it. When horizontal shifts are applied, the asymptote moves with the graph. For $f(x) = \log_b(x - h)$, the asymptote is at $x = h$.

X-intercept: For the parent function, $\log_b(x) = 0$ when $x = 1$ (since $b^0 = 1$ for any valid base). So the x-intercept is $(1, 0)$. For a horizontally shifted function $\log_b(x - h)$, the x-intercept shifts to $(1 + h, 0)$. If there's also a vertical shift, you'll need to solve $\log_b(x - h) + k = 0$ to find the new intercept.

Y-intercept: The parent function has no y-intercept because $\log_b(0)$ is undefined. Transformed functions can have a y-intercept if $x = 0$ falls within the domain. For instance, $f(x) = \log_2(x + 4)$ has a y-intercept at $f(0) = \log_2(4) = 2$.

End behavior (for $b > 1$):

1. As $x \to 0^+$, $f(x) \to -\infty$
2. As $x \to +\infty$, $f(x) \to +\infty$ (though the rate of increase slows dramatically)

Additional Properties of Logarithmic Functions

• Continuity: Logarithmic functions are continuous on their entire domain.
• Monotonicity: When $b > 1$, the function is strictly increasing. When $0 < b < 1$, the function is strictly decreasing. Either way, it's one-to-one, which is why logarithms are invertible.
• Concavity: For $b > 1$, logarithmic functions are always concave down on their domain. This means the graph rises quickly at first and then levels off, growing more and more slowly. For $0 < b < 1$, the function is concave up.