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

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

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