Graphing Logarithmic Functions: Properties, Transformations, and Examples

Graphing Logarithmic Functions

Introduction to Logarithmic and Exponential Functions

Logarithmic functions are the inverses of exponential functions. Understanding their graphs and properties is essential for solving equations and modeling real-world phenomena in Precalculus.

• Exponential Function: $f(x) = a^x$

• Logarithmic Function: $f(x) = \log_a x$

• The graph of $y = \log_a x$ is the reflection of $y = a^x$ across the line $y = x$.

Key Properties of Exponential and Logarithmic Functions

Exponential Functions	Logarithmic Functions
Form: $f(x) = a^x$ Domain: $(-\infty, \infty)$ Range: $(0, \infty)$ Horizontal asymptote: $y = 0$ Passes through $(0, 1)$	Form: $f(x) = \log_a x$ Domain: $(0, \infty)$ Range: $(-\infty, \infty)$ Vertical asymptote: $x = 0$ Passes through $(1, 0)$

Graphing Logarithmic Functions

• To graph $y = \log_a x$, plot key points such as $(1, 0)$, $(a, 1)$, and $(\frac{1}{a}, -1)$.

• The graph increases if $a > 1$ and decreases if $0 < a < 1$.

• Vertical asymptote at $x = 0$.

Example: Graph $f(x) = 3^x$ and $g(x) = \log_3 x$ on the same axes. The exponential function passes through $(0, 1)$, $(1, 3)$, $(-1, \frac{1}{3})$. The logarithmic function passes through $(1, 0)$, $(3, 1)$, $(\frac{1}{3}, -1)$.

Domain and Range

• Exponential: Domain $(-\infty, \infty)$, Range $(0, \infty)$

• Logarithmic: Domain $(0, \infty)$, Range $(-\infty, \infty)$

Transformations of Logarithmic Functions

Logarithmic graphs can be transformed using shifts, reflections, and stretches/compressions. The general form is:

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

• Horizontal shift: $h$ units right if $h > 0$, left if $h < 0$

• Vertical shift: $k$ units up if $k > 0$, down if $k < 0$

• Reflection: Over the $x$-axis if $a < 0$

• Vertical stretch/compression: By factor $|a|$

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

• Parent function: $\log_2 x$

• Shift right by 1, up by 2

• Vertical asymptote at $x = 1$

Practice Example

• Given $g(x) = -\log_2 (x + 1) + 2$

• Domain: $x > -1$

• Range: $(-\infty, \infty)$

• Vertical asymptote: $x = -1$

• Reflection over $x$-axis, shift left by 1, up by 2

Summary Table: Logarithmic Function Transformations

Transformation	Effect on Graph
$y = \log_a x$	Parent function
$y = \log_a (x - h)$	Shift right by $h$ units
$y = \log_a (x + h)$	Shift left by $h$ units
$y = \log_a x + k$	Shift up by $k$ units
$y = -\log_a x$	Reflect over $x$-axis

Additional info: Understanding these transformations is crucial for solving equations and modeling with logarithmic functions in Precalculus and beyond.