6.4 Graphs of Logarithmic Functions

Graphs of Logarithmic Functions

Logarithmic functions are the inverses of exponential functions, and their graphs have a distinctive shape you need to recognize. Understanding how to graph them, identify their key features, and apply transformations is central to this section.

Graphs of Logarithmic Functions

Domain and Range of Logarithms

The domain of a logarithmic function $f(x) = \log_b(x)$ is all positive real numbers: $(0, \infty)$. That's because the argument of the logarithm (the expression inside the $\log$) must be strictly greater than 0. You can't take the log of zero or a negative number.

The range is all real numbers: $(-\infty, \infty)$.

• As $x$ approaches 0 from the right, $\log_b(x)$ drops toward $-\infty$.
• As $x$ grows larger and larger, $\log_b(x)$ keeps increasing toward $+\infty$, though it does so slowly.

When transformations change the argument (like $\log_b(x - h)$), the domain shifts accordingly. For example, $f(x) = \log_2(x - 3)$ has a domain of $(3, \infty)$ because you need $x - 3 > 0$.

Graphing Logarithmic Transformations

The parent function is $f(x) = \log_b(x)$, where $b > 0$ and $b \neq 1$. The two most common bases are 10 (written $\log(x)$, the common log) and $e$ (written $\ln(x)$, the natural log).

Since logarithmic functions are inverses of exponential functions, the equation $y = \log_b(x)$ is equivalent to $b^y = x$. Graphically, this means the log curve is a reflection of the exponential curve $y = b^x$ over the line $y = x$. That's a useful way to visualize the shape if you already know what exponentials look like.

Transformations follow the same rules you've used for other function families:

• Vertical shift: $f(x) = \log_b(x) + k$ shifts the graph up $k$ units (down if $k < 0$).
• Horizontal shift: $f(x) = \log_b(x - h)$ shifts the graph right $h$ units (left if $h < 0$). This also moves the vertical asymptote to $x = h$.
• Vertical stretch/compression: $f(x) = a \cdot \log_b(x)$
  • Stretches vertically by a factor of $|a|$ when $|a| > 1$
  • Compresses vertically by a factor of $|a|$ when $0 < |a| < 1$
• Reflection over the x-axis: $f(x) = -\log_b(x)$ flips the graph upside down.

To graph a transformed log function step by step:

1. Identify the parent function and its base.
2. Determine the horizontal shift $h$ and vertical shift $k$.
3. Move the vertical asymptote from $x = 0$ to $x = h$.
4. Plot the transformed version of key reference points. For the parent function, convenient points are $(1, 0)$, $(b, 1)$, and $(1/b, -1)$. Apply each transformation to these coordinates.
5. Sketch the curve through your transformed points, approaching the asymptote on one side and rising slowly on the other.

Key Features of Logarithmic Graphs

Vertical asymptote: The parent function $f(x) = \log_b(x)$ has a vertical asymptote at $x = 0$. The graph gets infinitely close to this line but never touches or crosses it. Horizontal shifts move the asymptote; for $f(x) = \log_b(x - h)$, the asymptote sits at $x = h$.

x-intercept: Set $y = 0$ and solve. For the parent function, $\log_b(x) = 0$ gives $x = 1$, so the x-intercept is $(1, 0)$. Transformations shift this point. For example, $f(x) = \log_2(x - 3)$ has its x-intercept at $(4, 0)$ because $\log_2(4 - 3) = \log_2(1) = 0$.

No y-intercept for the parent function, since $x = 0$ is outside the domain (it's the asymptote). A horizontal shift can create a y-intercept, though. For instance, $f(x) = \log_2(x + 4)$ is defined at $x = 0$: $f(0) = \log_2(4) = 2$, giving a y-intercept of $(0, 2)$.

End behavior:

1. As $x \to 0^+$, $f(x) \to -\infty$ (the graph plunges downward near the asymptote).
2. As $x \to +\infty$, $f(x) \to +\infty$ (the graph rises without bound, but increasingly slowly).

This slow growth is a defining trait of logarithmic functions. Compare: $\log_{10}(10) = 1$, $\log_{10}(100) = 2$, $\log_{10}(1000) = 3$. The input has to multiply by 10 each time just to increase the output by 1.