Graphing Logarithmic Functions: Properties, Transformations, and Examples

Graphing Logarithmic Functions

Introduction to Logarithmic Functions

Logarithmic functions are the inverses of exponential functions and play a crucial role in Precalculus. Understanding their graphs, domains, ranges, and transformations is essential for analyzing their behavior and solving equations involving logarithms.

• Logarithmic Function: The general form is $f(x) = \log_b x$, where $b > 0$ and $b \neq 1$.

• Inverse Relationship: The logarithmic function $f(x) = \log_b x$ is the inverse of the exponential function $g(x) = b^x$.

• Graphing: The graph of $f(x) = \log_b x$ can be obtained by reflecting the graph of $g(x) = b^x$ over the line $y = x$.

Exponential vs. Logarithmic Functions

Exponential and logarithmic functions are closely related. Their graphs and properties can be compared as follows:

Exponential Functions	Logarithmic Functions
$f(x) = b^x$	$g(x) = \log_b x$
Domain: $(-\infty, \infty)$	Domain: $(0, \infty)$
Range: $(0, \infty)$	Range: $(-\infty, \infty)$
Horizontal asymptote: $y = 0$	Vertical asymptote: $x = 0$
Graph increases (if $b > 1$)	Graph increases (if $b > 1$)

Key Properties of Logarithmic Functions

• Domain: $x > 0$

• Range: $(-\infty, \infty)$

• Vertical Asymptote: $x = 0$

• Intercept: $y$-intercept at $(1, 0)$

• Increasing/Decreasing: For $b > 1$, the graph increases; for $0 < b < 1$, the graph decreases.

Example: Graphing $f(x) = 3^x$ and $g(x) = \log_3 x$

• Table of values for $f(x) = 3^x$ and $g(x) = \log_3 x$:

x	$f(x) = 3^x$	$g(x) = \log_3 x$
-2	1/9	-2
-1	1/3	-1
0	1	0
1	3	1
2	9	2

• Domain of $f(x) = 3^x$: $(-\infty, \infty)$

• Range of $f(x) = 3^x$: $(0, \infty)$

• Domain of $g(x) = \log_3 x$: $(0, \infty)$

• Range of $g(x) = \log_3 x$: $(-\infty, \infty)$

Transformations of Logarithmic Functions

Logarithmic functions can be transformed using rules similar to those for other functions. The general transformation is:

• Parent Function: $f(x) = \log_b x$

• Transformation: $g(x) = a \cdot f(c(x - h)) + k$

Common transformations include:

• Reflection: $-\log_b x$ reflects the graph over the $x$-axis.

• Horizontal Shift: $\log_b(x - h)$ shifts the graph $h$ units to the right.

• Vertical Shift: $\log_b x + k$ shifts the graph $k$ units up.

• Horizontal Stretch/Compression: $\log_b(cx)$ stretches or compresses the graph horizontally.

The general transformation formula is:

• $g(x) = f(c(x - h)) + k$

Example: Graphing $g(x) = \log_2(x - 1) + 2$

• Step 1: Graph the parent function $f(x) = \log_2 x$.

• Step 2: Shift the graph 1 unit to the right.

• Step 3: Shift the graph 2 units up.

• Domain: $x > 1$

• Range: $(-\infty, \infty)$

Practice Problem

• Graph $g(x) = -\log_2(x + 1) + 1$

• Step 1: Graph $f(x) = \log_2 x$.

• Step 2: Shift 1 unit left ($x + 1$).

• Step 3: Reflect over $x$-axis (negative sign).

• Step 4: Shift 1 unit up.

• Domain: $x > -1$

• Range: $(-\infty, \infty)$

Summary Table: Properties of Logarithmic Functions

Property	Logarithmic Function $f(x) = \log_b x$
Domain	$(0, \infty)$
Range	$(-\infty, \infty)$
Vertical Asymptote	$x = 0$
Intercept	$(1, 0)$
Increasing/Decreasing	Increasing if $b > 1$, decreasing if $0 < b < 1$

Additional info: These notes expand on the graphical and transformational properties of logarithmic functions, including step-by-step examples and practice problems for Precalculus students.