Graphing Logarithmic Functions and Transformations

Relationship Between Exponential and Logarithmic Functions

Logarithmic functions are mathematically defined as the inverses of exponential functions. To understand this relationship, consider the exponential function $f(x) = 3^x$. The table of values for this function reveals the following data points:

• In the domain $(-\infty, \infty)$, when $x = -2$, $y = \frac{1}{9}$.
• When $x = -1$, $y = \frac{1}{3}$.
• When $x = 0$, $y = 1$.
• When $x = 1$, $y = 3$.
• When $x = 2$, $y = 9$.

The range for the exponential function $f(x) = 3^x$ is $(0, \infty)$.

To graph the inverse function, denoted as $f^{-1}(x)$, you must swap the $x$ and $y$ values. This inverse relationship results in the equation $x = 3^y$, which is rewritten in logarithmic form as $y = \log_{3}(x)$. The points for this logarithmic function are:

• When $x = \frac{1}{9}$, $y = -2$.
• When $x = \frac{1}{3}$, $y = -1$.
• When $x = 1$, $y = 0$.
• When $x = 3$, $y = 1$.
• When $x = 9$, $y = 2$.

The domain and range of the logarithmic function are the exact opposites of the exponential function: the Domain is $(0, \infty)$ and the Range is $(-\infty, \infty)$.

Key Features Comparison

The following table summarizes the distinct characteristics of exponential parent functions versus logarithmic parent functions:

1. Function $f(x) = b^x$ (Exponential)    - Domain: $(-\infty, \infty)$    - Range: $(0, \infty)$    - Asymptote: $y = 0$ (Horizontal Asymptote)    - Intercept: $y\text{-intercept at } (0, 1)$

2. Function $f(x) = \log_{b}(x)$ (Logarithmic)    - Domain: $(0, \infty)$    - Range: $(-\infty, \infty)$    - Asymptote: $x = 0$ (Vertical Asymptote)    - Intercept: $x\text{-intercept at } (1, 0)$

Method for Graphing Logarithmic Functions

To graph a logarithmic function such as $y = \log_{4}(x)$, identifying the base and distinct coordinates is essential.

• Base Identification: In the function $y = \log_{4}(x)$, the base $b = 4$.
• Three Distinct Points: Every logarithmic parent function of the form $y = \log_{b}(x)$ passes through three predictable points:    1. $(\frac{1}{b}, -1)$: For this example, $(\frac{1}{4}, -1)$.    2. $(1, 0)$: This is the universal $x\text{-intercept}$.    3. $(b, 1)$: For this example, $(4, 1)$.

Once these points are plotted, connect them with a smooth curve that approaches but never touches the vertical asymptote. For $y = \log_{4}(x)$, the Domain is $(0, \infty)$ and the Range is $(-\infty, \infty)$.

Logarithmic Transformations

The general transformed logarithmic function is represented by the equation $y = a \log_{b}(x - h) + k$. Each variable represents a specific transformation of the parent graph:

• $|a| \geq 1$: Indicates a vertical stretch.
• $0 < |a| < 1$: Indicates a vertical shrink.
• $a < 0$: If the value of $a$ is negative, the graph is reflected over the $x\text{-axis}$.
• $h$: This represents a horizontal translation. A positive value shifts the graph right, and a negative value shifts it left (note the formula uses $x - h$).
• $k$: This represents a vertical translation. A positive value shifts the graph up, and a negative shifts it down.

The Order of Transformations (HSRV)

When graphing transformations, it is vital to follow a specific sequence known as HSRV:

1. H (Horizontal): Horizontal shifts.
2. S (Stretch/Shrink): Vertical stretching or shrinking.
3. R (Reflect): Reflections over the axes.
4. V (Vertical): Vertical shifts.

Example Graphing with Transformations

Consider the function $y = 2 \log_{2}(x - 2) + 3$. The parent function is $y = \log_{2}(x)$.

Parent Points for $y = \log_{2}(x)$:

• $(\frac{1}{2}, -1)$
• $(1, 0)$
• $(2, 1)$

Identified Transformations:

• $h = 2$: Shifts the graph right 2 units.
• $k = 3$: Shifts the graph up 3 units.
• $a = 2$: Indicates a vertical stretch by a factor of 2 (no reflection).

Point Transformation Process:

1. Horizontal Shift ($x + 2$):    - $(\frac{1}{2} + 2, -1) = (2.5, -1)$    - $(1 + 2, 0) = (3, 0)$    - $(2 + 2, 1) = (4, 1)$

2. Vertical Stretch and Shift ($2y + 3$):    - For point $(2.5, -1)$: $2(-1) + 3 = 1$. Final point: $(2.5, 1)$.    - For point $(3, 0)$: $2(0) + 3 = 3$. Final point: $(3, 3)$.    - For point $(4, 1)$: $2(1) + 3 = 5$. Final point: $(4, 5)$.

Guided Practice Example

Graph the function $y = -\log_{10}(x + 3) - 1$.

Base Identification and Parent Points: The base is 10. The parent function points are $(\frac{1}{10}, -1)$, $(1, 0)$, and $(10, 1)$.

Transformations:

• $h = -3$: Horizontal shift left 3 units.
• $k = -1$: Vertical shift down 1 unit.
• $a = -1$: Reflection over the $x\text{-axis}$.

Transformed Points Calculation: Using the sequence $x - 3$ and $-y - 1$:

• Point 1: $(\frac{1}{10} - 3, -(-1) - 1) = (-2.9, 0)$
• Point 2: $(1 - 3, -(0) - 1) = (-2, -1)$
• Point 3: $(10 - 3, -(1) - 1) = (7, -2)$

The Final Product points are $(-2.9, 0)$, $(-2, -1)$, and $(7, -2)$.

Writing Logarithmic Functions from Graphs

You can determine the equation of a logarithmic function by identifying its translations relative to a parent function.

Example 8: Analyzing Translations If a graph is based on $f(x) = \log_{4}(x)$ but has been translated 5 units up, then the value of $k$ is 5. The resulting function is $g(x) = \log_{4}(x) + 5$.

Case Studies for $f(x) = \log_{3}(x)$: a. Vertical Translation: For a graph undergoing a vertical shift $g(x) = f(x) + k$, if the graph is shifted up 4 units, the equation becomes $g(x) = \log_{3}(x) + 4$. b. Vertical Stretch/Shrink: For a graph undergoing a compression via $g(x) = k \cdot f(x)$, where the points are shrunk vertically by a factor of $\frac{1}{3}$, the resulting equation is $g(x) = \frac{1}{3} \log_{3}(x)$.