Logarithmic Function Graphs

Logarithmic functions are the inverses of exponential functions. Understanding how to graph them and apply transformations is a fundamental skill in algebra and pre-calculus.

The Parent Graph: $y = \log_b(x)$

The most basic logarithmic function is $y = \log_b(x)$, where the base $b > 0$ and $b \neq 1$. No matter what base you use, the parent graph shares several key properties:

• Domain: $x > 0$. You can only take the logarithm of positive numbers.
• Range: All real numbers ($-\infty < y < \infty$).
• x-intercept: The graph always passes through $(1, 0)$ because $\log_b(1) = 0$.
• Key Point: The graph passes through $(b, 1)$ because $\log_b(b) = 1$.
• Vertical Asymptote: The y-axis, or the line $x = 0$. As $x$ gets closer to $0$ from the right, the graph plunges downward toward $-\infty$ (for $b > 1$).

Comparing Different Bases

What happens when we change the base? Let's compare $y = \log_2(x)$ and $y = \log_5(x)$.

Both graphs pass through $(1, 0)$ and have a vertical asymptote at $x = 0$. However, they differ in how steeply they curve:

• For $x > 1$, $y = \log_5(x)$ grows slower than $y = \log_2(x)$. For example, to reach a height of $y = 2$, the base-2 graph only needs $x = 4$, but the base-5 graph needs $x = 25$.
• For $0 < x < 1$, $y = \log_5(x)$ is closer to the y-axis and x-axis than $y = \log_2(x)$.

Transformations of Logarithmic Graphs

You can move, stretch, and flip logarithmic graphs using the standard transformation formula:

$y = a\log_b(x - h) + k$

• $h$ (Horizontal Shift): Moves the graph left or right. This changes the domain and shifts the vertical asymptote to $x = h$.
• $k$ (Vertical Shift): Moves the graph up or down.
• $a$ (Vertical Stretch/Compression): Stretches the graph vertically. If $a$ is negative, it reflects the graph across the x-axis.

Example: Graphing with Transformations

Problem: Graph $y = \log_2(x - 3) + 1$ and identify the domain and asymptote.

Solution:

1. Identify the parent function: The parent function is $y = \log_2(x)$, which has a vertical asymptote at $x = 0$ and key points at $(1,0)$ and $(2,1)$.
2. Apply the horizontal shift: The $(x - 3)$ inside the logarithm shifts the entire graph right by 3 units.
   • The new vertical asymptote is $x = 3$.
   • The domain is found by setting the argument greater than zero: $x - 3 > 0$, so the domain is $x > 3$.
3. Apply the vertical shift: The $+ 1$ on the outside shifts the graph up by 1 unit.
4. Find new key points:
   • Original $(1,0) \rightarrow$ shift right 3 to $(4,0) \rightarrow$ shift up 1 to $(4,1)$.
   • Original $(2,1) \rightarrow$ shift right 3 to $(5,1) \rightarrow$ shift up 1 to $(5,2)$.

To sketch the graph, draw a dashed vertical line at $x = 3$ for the asymptote. Plot the points $(4,1)$ and $(5,2)$, and draw a smooth curve that passes through these points and approaches $x = 3$ as it moves downward.